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LIBRARY 

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HENRY         HOLT         AND         COMPANY 

NEW  YORK  CHICAGO 


(FUNCTIONS 

OF  A 

COMPLEX  VAEIABLE 


BY 


E.  J.  TOWNSEND,  Ph.D. 

i'/ 

PROFESSOR    OF    MATHEMATICS,    UNIVERSITY    OF    ILLINOIS 


NEW   YORK 

HENRY  HOLT   AND   COMPANY 

1915 


Copyright,  1915, 

BY 

HENRY  HOLT  AND  COMPANY 


Stanbope  iprcss 

F.   H.  GILSON   COMPANY 
BOSTON,  U.S.A. 


PREFACE 

The  present  volume  is  based  on  a  course  of  lectures  given  by  the 
author  for  a  number  of  years  at  the  University  of  Illinois.  It  is  in- 
tended as  an  introductory  course  suitable  for  first  year  graduate 
students  and  assumes  a  knowledge  of  only  such  fundamental  prin- 
ciples of  analysis  as  the  student  will  have  had  upon  completing  the 
usual  first  course  in  calculus.  Such  additional  information  concern- 
ing functions  of  real  variables  as  is  needed  in  the  development  of 
the  subject  has  been  introduced  as  a  regular  part  of  the  text.  Thus 
a  discussion  of  the  general  properties  of  line-integrals,  a  proof  of 
Green's  theorem,  etc.,  have  been  included.  The  material  chosen 
deals  for  the  most  part  with  the  general  properties  of  functions  of  a 
complex  variable,  and  but  little  is  said  concerning  the  properties  of 
some  of  the  more  special  classes  of  functions,  as  for  example  elliptic 
functions,  etc.,  since  in  a  first  course  these  subjects  can  hardly  be 
treated  in  a  satisfactory  manner. 

The  course  presupposes  no  previous  knowledge  of  complex  numbers 
and  the  order  of  development  is  much  as  that  commonly  followed  in 
the  calculus  of  real  variables.  Integration  is  introduced  early,  in 
connection  with  differentiation.  In  fact  the  first  statement  of  the 
necessary  and  sufficient  condition  that  a  function  is  holomorphic  in 
a  given  region  is  made  in  terms  of  an  integral.  By  this  order  of 
arrangement,  it  is  possible  to  establish  early  in  the  course  the  fact 
that  the  continuity  of  the  derivative  follows  from  its  existence,  and 
consequently  the  Cauchy-Goursat  and  allied  theorems  can  be  dem- 
onstrated without  any  assumption  as  to  such  continuity.  Likewise, 
it  can  thus  be  shown  that  Laplace's  differential  equation  is  satisfied 
without  making  the  usual  assumptions  as  to  the  existence  of  the 
derivatives  of  second  order.  The  term  holomorphic,  often  omitted, 
has  been  used  as  expressing  an  important  property  of  single-valued 
functions,  reserving  the  use  of  the  term  analytic  for  use  in  connection 
with  functions  derived  from  a  given  element  by  means  of  analytic 
continuation.  While  the  Cauchy-Riemann  viewpoint  is  that  first 
iatroduced,  attention  is  called  to  the  Weirstrass  development  in  the 


IV  PREFACE 

chapter  on  series,  and  in  subsequent  discussions  either  definition  of 
an  analytic  function  is  used  as  best  suits  the  purpose  in  hand. 

In  Chapter  IV  much  use  is  made  of  mapping,  thus  enabling  us  to 
consider  in  connection  with  the  definition  of  certain  elementary 
functions  some  of  their  more  important  uses  in  physics.  For  the 
same  reason  in  Chapter  V  the  consideration  of  linear  fractional 
transformation  is  especially  emphasized  and  discussed  as  a  kinematic 
problem.  The  discussion  of  series  in  Chapter  VI  lays  the  foundation 
for  the  consideration  of  the  fundamental  properties  of  single-valued 
functions  discussed  in  the  following  chapter.  In  the  final  chapter, 
it  is  pointed  out  how  these  properties  may  be  extended  to  the  con- 
sideration of  multiple-valued  functions. 

The  author  wishes  to  express  his  appreciation  of  the  helpful  sug- 
gestions which  have  been  given  to  him  by  Professor  J.  L.  Markley  of 
the  University  of  Michigan,  Professor  A.  Dresden  of  the  University 
of  Wisconsin,  Professor  W.  A.  Hurwitz  of  Cornell  University,  and  to 
Dr.  Otto  Dunkel  of  the  University  of  Missouri,  who  have  read  the 
proof  sheets.  He  is  also  under  obligations  to  his  colleagues  Dr. 
Denton  and  Dr.  Kempner,  who  have  read  the  manuscript.  Finally, 
he  wishes  to  express  especially  his  obligations  to  Dr.  George  Rut- 
ledge,  who  has  rendered  him  valuable  assistance  in  the  preparation 
of  the  manuscript. 

E.  J.  TOWNSEND. 

University  of  Illinois 
July,  1915 


ERRATA. 


Page    26,  line  8,  insert  5  after  number. 

"  27,  line  5  from  bottom,  for  (4)  read  (8). 

"  38,  replace  Zo  by  U  and  zi  by  ti. 

"  "  ,  line  16  from  bottom,  iovf{z)  read  f{t). 

"  55,  Eq.  (1),  for  J  read    P*. 

"  71,  Fig.  26,  forCi'readC* 

"  88,  line  7,  for /(a:)  read/(0). 

"  "  ,  line  6  from  bottom,  for  Ax  read  Az. 

"  92,  line  11  from  bottom,  for    I    read    /    . 

"  93,  line  15,  for  (2)  read  (3). 

"  101,  line  8  from  bottom,  for  2  ty  read  2  ixy. 

"  104,  line  4  from  bottom,  for  Art.  21  read  Art.  22. 

"  106,  Fig.  36,  interchange  V  and  U. 

"  112,  line  6  from  bottom,  for  4  w  —  4  read  4  m  +  4. 

"  155,  Ex.  20,  for  log  '^^^  read  log  ^  "  !  ' 
w  —  1                 w-\-\ 

"  178,  line  2  from  bottom,  for  Z4  read  Zt. 

"  195,  lines  16-18,  replace  P  by  Pi,  and  "  at  infinity  "  by  P. 

"  240,  line  15,  for  C  read  Ci. 

"  277,  line  8  from  bottom,  for  /    read    /  . 

Jc  JCt 

"  286,  line  14,  transpose  I    and  "  when  ". 

"  299,  line  7  from  bottom,  for  Z\  read  z'^. 

"  340,  Eq.  (6),  for  e"'('"^^)  read  e"'^'^"^). 

"  "  ,  line  8  from  bottom,  for  f  0,  ^j  read  ( ^,  0 j. 

"  344,  line  9,  for  tz  read  T3. 

"  363,  hne  3  from  bottom,  for  k  read  A;  +  1. 

read    I     . 


M 


CONTENTS 

CHAPTER  I 
REAL  AND  COMPLEX  NUMBERS 

ARTICLE  PAGE 

1.  Rational  Numbers 1 

2.  Irrational  Numbers 2 

3.  System  of  Real  Numbers 4 

4.  Complex  Numbers 5 

5.  Geometric  Representation  of  Complex  Numbers 6 

6.  Comparison  of  Complex  Numbers 8 

7.  Addition  and  Subtraction  of  Complex  Nimibers 8 

8.  Multiplication  of  Complex  Numbers 11 

9.  Division  of  Complex  Numbers 16 


CHAPTER  II 
FUNDAMENTAL  DEFINITIONS  CONCERNING  FUNCTIONS 

10.  Constants,  Variables 20 

11.  Definition  and  Classification  of  Functions 21 

12.  Limits 23 

13.  Continuity 33 

CHAPTER  III 

DIFFERENTIATION  AND  INTEGRATION 

14.  Differentiation;  Definition  of  an  Analytic  Function 43 

15.  Line-integrals 46 

16.  Green's  Theorem 54 

17.  Integral  of  /  (z) .". . .  60 

18.  Change  of  Variable,  Complex  to  Real 64 

19.  Cauchy-Goursat  Theorem 66 

20.  Cauchy's  Integral  Formula 75 

21.  Cauchy-Riemann  Differential  Equations 82 

22.  Change  of  Complex  Variable 89 

23.  Indefinite  Integrals 90 

24.  Laplace's  Differential  Equation 92 

25.  Applications  to  Physics 96 

V 


VI  CONTENTS 

CHAPTER  IV 
MAPPING,  WITH  APPLICATIONS  TO  ELEMENTARY  FUNCTIONS 

ARTICLE  PAOB 

26.  Conjugate  Functions 101 

27.  Conformal  Mapping 104 

28.  The  Function  w  =  z" 114 

29.  Definition  and  Properties  of  e' 122 

30.  The  Function  w  =  logz 133 

31.  Trigonometric  Functions 144 

32.  Hyperbolic  Functions 150 

CHAPTER  V 
LINEAR  FRACTIONAL  TRANSFORMATIONS 

S3.   Definition  of  Linear  Fractional  Transformations 156 

34.  Point  at  Infinity 157 

35.  The  Transformation  u>  =  2  +  /3 159 

36.  The  Transformation  w  =  az 159 

37.  The  Transformation  w  —  az-\-0 162 

38.  The  Transformation  w=  - 166 

z 

39.  General  Properties  of  the  Transformation  w  =  f— ; 173 

72  +  5 

40.  Stereographic  Projection 184 

41.  Classification  of  Linear  Fractional  Transformations 190 

CHAPTER  VI 
INFINITE  SERIES 

42.  Series  with  Complex  Terms 198 

43.  Operations  with  Series 206 

44.  Double  Series 213 

45.  Uniform  Convergence 217 

46.  Integration  and  Differentiation  of  Series 222 

47.  Power  Series 226 

48.  Expansion  of  a  Function  in  a  Power  Series 238 

CHAPTER  VII 
GENERAL  PROPERTIES  OF  SINGLE-VALUED  FUNCTIONS 

49.  Analytic  Continuation 245 

60.  Definition  of  Analytic  Function 257 

61.  Singular  Points,  Zero  Points 262 

62.  Laurent's  Expansion 275 

63.  Residues 284: 

64.  Rational  Fimctions,  Fundamental  Theorem  of  Algebra 290 


CONTENTS  Vll 

ARTICLE  PAGE 

66.   Transcendental  Functions 300 

66.  Mittag-Leffler's  Theorem 303 

67.  Expansion  of  Functions  by  Infinite  Products 308 

68.  Periodic  Functions 317 

CHAPTER  VIII 
MULTIPLE-VALUED  FUNCTIONS 

69.  Fundamental  Definitions 329 

60.  Biemann  Surface  iorvfi— 3w  —  2z=0 338 

. T^~ 

61.  Riemann  Surface  for  w  =  Vz  —  Zo  +  v 343 

y  z  —  zi 

62.  Riemann  Surface  for  w  =  log  z 346 

63.  Branch-points,  Branch-cuts 347 

64.  Stereographic  Projection  of  a  Riemann  Surface 354 

66.    General  Properties  of  Riemann  Surfaces 355 

66.  Singular  Points  of  Multiple- valued  Functions 358 

67.  Functions  Defined  on  a  Riemann  Surface.     Physical  Applications ....  362 

68.  Function  of  a  Function 367 

69.  Algebraic  Functions 368 

Index i 381 


FUNCTIONS  OF  A  COMPLEX  VARIABLE 


CHAPTER  I 
REAL  AND   COMPLEX   NUMBERS 

1.  Rational  numbers.  Some  understanding  of  the  nature  of  a 
number,  the  classes  into  which  numbers  may  be  divided,  and  the 
general  laws  governing  the  fundamental  operations  with  them  is 
essential  to  the  study  of  the  theory  of  functions.  We  obtain  our 
first  notion  of  numbers  when  we  undertake  to  enumerate  the  indi- 
viduals composing  a  group  of  objects.  The  process  of  counting 
leads,  however,  only  to  the  positive  integers.  We  arrive  at  the 
same  result  when  we  assume  the  existence  of  unity  and  a  certain 
mathematical  process  known  as  addition.  Furthermore,  the  posi- 
tive integers  obey  the  following  law: 

Given  any  two  positive  integers  a  and  b{b  >  a),  there  exists  one  and 
only  one  positive  integer  x  such  that 

a  -\-  X  =  b. 

It  becomes  at  once  apparent  that  the  positive  integers  do  not 
completely  serve  the  purpose  of  analysis  when  we  attempt  to  solve 
the  above  equation  for  the  case  where  a  =  b.  In  order  to  give  any 
interpretation  at  all  to  the  solution  in  this  case,  it  is  necessary  to 
introduce  a  new  number  called  zero,  defined  by  the  identity 

a  -\-  0  =  a. 

If  a  is  allowed  to  be  greater  than  b,  it  is  again  necessary  to  ex- 
tend the  domain  of  the  number-system  by  the  introduction  of  nega- 
tive numbers  in  order  to  give  an  interpretation  to  the  solution  of 
the  above  equation.  Even  with  this  extension  of  the  number- 
system,  it  is  impossible  to  solve  all  linear  equations.  Suppose,  for 
example,  it  is  required  to  find  the  value  of  x  from  the  equation 

ax  =  b,         a  9^  0. 

A  number-system  that  includes  only  positive  and  negative  integers  is 
inadequate  to  interpret  the  result 

6 

a 
1 


2  REAL  AND  COMPLEX  NUMBERS  [Chap.  L 

whenever  b  is  not  an  integral  multiple  of  a.  A  further  extension 
of  the  number-system  now  becomes  necessary  and  this  extension  is 
gained  by  the  introduction  of  fractions. 

The  numbers  thus  far  discussed,  that  is  integers  including  zero, 
and  fractions,  constitute  a  system  of  numbers  called  rational  num- 
bers.*   A  characteristic  property  of  such  numbers  is  that  they  may 

always  be  expressed  in  the  form  - ,  where  a  and  h  are  integers  prime 

CL 

to  each  other  and  a^  0.  By  the  aid  of  the  symbols  for  the  funda- 
mental operations  of  arithmetic  rational  numbers  can  always  be 
expressed  by  a  finite  number  of  digits.  It  is  possible  and  often  con-  • 
venient  to  express  such  numbers  by  means  of  an  infinite  sequence 
of  digits,  but  it  is  not  necessary  to  do  so.  Thus  f  is  a  rational 
number,  but  when  expressed  in  the  form  of  a  decimal  fraction  we 
have 

i  =  0.3333  .... 

2.  Irrational  numbers.  If  we  undertake  to  solve  equations  of  a 
higher  degree  than  the  first,  the  system  of  rational  numbers  often' 
proves  insufl&cient.     For  example,  if  we  have  given  the  equation 

a:2  -  2  =  0 

to  find  the  value  of  x,  we  have  a;  =  ±  v^,  a  result  that  has  no 
interpretation  in  the  domain  of  rational  numbers.     To  show  that 

no  such  interpretation  is  possible,  assume  r  =  ±  '^2,  a  and  b  being 

integers  prime  to  each  other.    We  have  then 

g  =  2,         a^  =  2b\ 

The  number  2  is  then  a  factor  of  a^  and  as  all  prime  factors  appear 
an  even  number  of  times  in  a  perfect  square,  2  must  appear  an 
even  number  of  times  in  a^.  Consequently,  2  must  also  appear  as  a 
factor  of  2  b^  an  even  number  of  times.  This,  however,  is  impos- 
sible, as  it  must  then  appear  as  a  factor  of  b^  itself  and  indeed  an 
even  number  of  times.  As  2  cannot  be  a  factor  of  one  member  of 
the  identity  an  even  number  of  times  and  of  the  other  an  odd  num- 
ber of  times,  the  assumption  that  V2  is  a  rational  number  is  not 
valid. 

*  For  a  more  complete  discussion  of  rational  numbers  the  reader  is  referred  to 
Pierpont,  Theory  of  Functions  of  Real  Variables,  Vol.  I,  Chap.  I. 


Art.  2.]  IRRATIONAL   NUMBERS  3 

We  shall  see  later  that  it  is  characteristic  of  a  new  class  of  num- 
bers, called  irrational  numbers  to  distinguish  them  from  the  num- 
bers discussed  in  the  preceding  article,  that  they  do  not  admit  of 

expression  in  the  form  j-  • 

To  see  more  clearly  the  nature  of  irrational  numbers,  let  us  con- 
sider the  totality  of  rational  numbers.  Suppose  we  separate  these 
numbers  into  two  sets  such  that  each  number  of  the  first  set  is 
greater  than  every  number  of  the  second  set.  Such  a  separation  of 
the  rational  system  of  numbers  is  called  a  partition.*  "We  have, 
for  example,  a  partition  if  we  select  any  rational  number  a  and  put 
into  one  set  Ai  all  those  rational  numbers  that  are  equal  to  or  greater 
than  a  and  into  a  second  set  A^  all  rational  numbers  that  are  less 
than  a.    In  this  case  the  number  a  is  itself  an  element  of  the  set  Ai. 

We  may  likewise  establish  a  partition  by  putting  into  the  set  Ai 
all  of  those  rational  numbers  greater  than  a  and  into  A2  all  those 
equal  to  or  less  than  a.  In  this  case  the  number  a  belongs  to  set  -A  2. 
It  will  be  noticed  that  by  the  first  partition  there  is  a  smallest  number 
in  Ai  and  by  the  second  partition  there  is  a  largest  number  in  A2. 
In  each  case  this  number  is  the  rational  number  a  itseK. 

It  is  possible,  however,  to  establish  a  partition  of  the  entire  sys- 
tem of  rational  numbers  in  such  a  manner  that  in  the  one  set  Ai 
there  shall  be  no  smallest  number  and  at  the  same  time  in  the  sec- 
ond set  A2,  there  shall  be  no  largest  number.  For  example,  let  us 
consider  again  V2,  As  we  have  seen,  this  number  is  not  a  rational 
number.  Put  into  set  Ai  all  of  those  rational  numbers  whose  squares 
are  greater  than  2  and  into  A2  all  rational  numbers  whose  squares 
are  less  than  2.  The  two  sets  Ai  and  A2  then  fulfill  the  conditions 
that  each  number  of  A 1  is  greater  than  any  number  of  A2  and  there 
is  no  smallest  number  in  Ai  and  no  largest  number  in  A2;  for,  no 
matter  how  near  to  2  the  square  of  a  particular  rational  number 
may  be,  there  are  always  other  rational  numbers  whose  squares  lie 
between  the  square  of  the  one  selected  and  2. 

The  notion  of  the  partition  of  the  system  of  rational  numbers 
affords  a  convenient  means  of  defining  irrational  numbers.  For  this 
purpose  suppose  the  totality  of  rational  numbers  to  be  divided  in 
any  manner  whatever  into  two  groups  Ai,  A2  having  the  following 
properties: 

*  Introduced  by  Dedekind,  Stetigkeit  und  irrationale  Zahlen,  Braunschweig, 
1872. 


4  REAL  AND  COMPLEX   NUMBERS  [Chap.  I. 

(1)  Each  number  of  the  set  Ai  shall  be  greater  than  any  number 
of  the  set  A^. 

(2)  There  shall  be  no  smallest  number  in  Ai  and  no  largest  number 
in  Ai. 

In  the  case  where  a  was  the  smallest  rational  number  in  Ai  or 
the  largest  one  in  A 2,  it  could  be  said  that  the  partition  defined 
uniquely  the  rational  number  a.  In  the  present  case,  it  can  no 
longer  be  said  that  the  partition  defines  a  rational  number;  for, 
every  rational  number  belongs  either  to  set  Ai  or  set  A^,  and  since 
by  (2)  there  can  be  no  smallest  number  in  Aj  and  no  largest  one  in 
Ai,  the  partition  can  not  define  a  number  in  either  set.  Conse- 
quently, the  partition  may  be  said  to  define  a  new  number;  we  call 
such  a  number  an  irrational  number.  The  fundamental  operations 
of  arithmetic  may  be  defined  for  irrational  numbers  in  a  manner 
consistent  with  the  corresponding  definitions  for  rational  numbers.* 

3.  System  of  real  numbers.  The  rational  numbers  and  the  irra- 
tional numbers  taken  together  constitute  a  system  of  numbers  known 
as  real  numbers.  It  is  this  system  of  numbers  that  lies  at  the  basis 
of  the  calculus  of  real  variables.  This  system  constitutes  a  closed 
group  with  respect  to  the  fundamental  operations  of  arithmetic  and 
obeys  certain  laws  already  familiar  to  the  student  from  his  study  of 
algebra.  For  any  numbers  a,  h,  c  of  this  system,  we  have  from  the 
definitions  of  those  fundamental  operations 

I.   For  addition: 

(1)  The  commutative  law:  a  +  6  =  6  +  a. 

(2)  The  associative  law :  a  +  (6  +  c)  =  (a  +  6)  -}-  c. 

II.   For  multiplication: 

(1)  The  commutative  law:  ab  =  ba. 

(2)  The  associative  law :  a(hc)  =  {ah)  c. 

(3)  The  distributive  law :  (a  +  6)  c  =  oc  +  &c. 

(4)  Factor  law:  If  ah  =  0,  then  a  =  0  or  6  =  0. 

It  is  customary  to  introduce  subtraction  and  division  as  the  in- 
verse operations  of  addition  and  multiplication.  From  the  defini- 
tion of  these  inverse  operations  and  the  foregoing  fundamental  laws 
follow,  as  purely  formal  consequences,  all  of  the  rules  of  operation 
for  real  numbers,  f 

*  See  Fine,  The  Number-System  of  Algebra,  Art.  29. 
t  Ibid.,  Arts.  10,  18. 


^ 


Art.  4.]  REAL  NUMBERS,   COMPLEX   NUMBERS  5 

We  assume  the  existence  of  a  one-to-one  correspondence  between 
the  totaUty  of  real  numbers  and  the  points  on  a  straight  line;  that 
is  to  say,  we  assume  that  to  each  real  number  can  be  assigned  a 
definite  point  on  the  line  and  conversely  to  every  such  point  there 
may  be  assigned  one  and  only  one  real  number.*  This  assumption 
makes  possible  a  geometric  interpretation  of  the  results  of  our  dis- 
cussion and  the  applications  of  analysis  to  geometry.  ,— — ■ 

4.  Complex  numbers.  It  will  be  observed  that  all  real  numbers 
arise  from  the  assumption  of  a  single  unit,  namely  1.  By  assuming 
the  additional  fundamental  unit  v  —  1,  which  we  shall  represent 
by  i,  a  very  important  extension  of  the  number-system  thus  far 
discussed  can  be  made.  By  the  use  of  these  two  units,  1  and  ^,  we 
can  construct  the  numbers  of  the  type  a-{-  ib,  where  a  and  b  are  real 
numbers.  It  becomes  necessary  to  extend  the  number-system  so  as 
to  include  numbers  of  this  type  if  the  solution  of  the  equation 

ax~  -\-  bx  -\-  c  =  0, 

where  ¥  —  4  ac  <  0,  is  to  have  any  meaning.  Such  numbers  are 
called  complex  numbers  and  form  the  basis  of  that  special  branch  of 
the  theory  of  functions  to  be  considered  in  this  volume.  It  will  be 
seen  that  since  a  and  b  may  take  all  real  values,  therefore  including 
zero,  real  numbers  are  a  special  case  of  complex  numbers,  that 
is,  complex  numbers  where  6  =  0.  In  considering  the  arithmetic 
of  complex  numbers,  the  question  arises  as  to  what  is  to  be  under- 
stood by  such  terms  as  "equal  to,"  "greater  than,"  etc.,  and  by 
the  fundamental  operations  of  addition,  subtraction,  etc.  More- 
over, it  cannot  be  assumed  in  advance  that  the  laws  of  operation 
with  real  numbers  may  be  extended  without  qualification  to  this 
broader  field.  Since  real  numbers  appear  as  a  special  case  of  com- 
plex numbers,  it  is  necessary  to  define  these  expressions  and  the 
fundamental  operations  in  such  a  manner  that  the  corresponding 
relations  between  real  numbers  shall  appear  as  special  cases.  These 
definitions  will  be  considered  in  the  following  articles. 

Complex  numbers  involving  more  than  two  units  have  been  used 
by  mathematicians.  For  example,  Hamilton,  a  distinguished  Eng- 
Hsh  mathematician,  introduced  higher  complex  numbers  known  as 

*  For  references  to  the  mathematical  literature  where  this  subject  is  discussed 
see:  Encyclopedie  des  Sciences  Mathematiques,  Tome  I,  Vol.  I,  pp.  146-147,  or 
Staude's  Analytische  Geometrie  des  Punktes,  der  geraden  Linie,  und  der  Ebene, 
p.  422  (10). 


6  REAL  AND  COMPLEX  NUMBERS  IChap.  I. 

quaternions.    For  this  purpose,  he  made  use  of  the  unit  1  and  the 
additional  units  i,  j,  k,  connected  by  the  following  relations: 
l2=  j2  =  fc2  =  ijk  =  -1. 

No  use  will  be  made  of  quaternions  or  of  other  higher  complex 
numbers  in  this  volume,  and  the  subject  is  mentioned  merely  to 
illustrate  the  possibility  of  further  extensions  of  the  number  concept. 

6.  Geometric  representation  of  complex  numbers.  The  as- 
sumption which  we  have  made  as  to  the  one-to-one  correspondence 
between  points  on  a  straight  Une  and  the  totality  of  real  numbers, 
makes  it  possible  to  give  a  geometric  representation  to  complex 
numbers.  For  this  purpose,  we  introduce  a  system  of  rectangular 
coordinates  similar  to  those  used  in  Cartesian  geometry. 

To  represent  the  number  *  a  -{-  ih,  lay  off  on  OX,  called  the  axis 
of  reals,  the  distance  a  and  on  OY,  called  the  axis  of  imaginaries, 
the  distance  6.  Draw  through  A  a  line  parallel  to  OF  and  through 
B  a  line  parallel  to  OX.  The  intersection  P  of  these  Unes  represents 
the  complex  number  a  -\-  ib.  The  numbers  a  and  h  may  be  any 
real  numbers,  positive  or  negative.  From  these  considerations,  it 
follows  that  there  exists  a  one-to-one  correspondence  between  the 
points  of  the  plane  and  the  totaUty  of  complex  numbers.  We  shall 
refer  to  the  plane,  used  in  this  way,  as  the  complex  plane.  From 
the  relation  between  the  points  of  the  complex  plane  and  the  totahty 
of  complex  numbers,  it  follows  that  the  complex  numbers  constitute 
a  continuous  system. 

By  making  use  of  the  trigonometric  functions,  it  is  possible  and 
frequently  convenient  to  represent  complex  numbers  in  another 
form.     From  Fig.  1,  we  have 

a  =  p  cos  6,        h  =  psind. 

We  may  therefore  write 

o  4-  i6  =  p(cos  0  +  t  sin  6). 

The  distance  OP  =  p  is  called  the  modulus  of  the  complex  num- 
ber, and  the  angle  6  is  called  the  amplitude  of  the  complex  number. 

*  The  first  mathematician  to  propose  a  geometric  interpretation  of  the  imagi- 
nary number  V-l  was  Kuhn  of  Danzig  in  1750-1751.  The  idea  was  extended 
by  Argand  in  1806  to  include  a  representation  of  complex  niunbers  of  the  form 
a  +  b  V  — 1,  a  representation  that  was  later  used  by  Gauss.  The  complex  plane 
is  frequently  referred  to  as  the  Argand  plane  or  the  Gauss  plane. 


Art.  5.1 


GEOMETRIC  REPRESENTATION 


P.(g+ib) 


Fig 


It  will  be  observed  that  for  any  given  number  a  -\-  ib  the  modulus  p 
is  a  single-valued  function  of  the  real  numbers  a  and  b,  while  the 
amplitude  9  is  a  multiple-valued  function  of  these  numbers.  The 
number  a^  -{-l^  =  p-  is  frequently  y. 
referred  to  as  the  norm  of  the  com- 
plex number  a  +  ib.  The  value  of 
6  lying  in  the  interval  —  ir  <  6  =  t 
is  called  the  chief  amplitude.  The 
amplitude  is  measured  positively 
in  a  counter-clockwise  direction. 
The  modulus  is  always  to  be  con- 
sidered as  positive,  and  hence  is 
often  referred  to  as  the  absolute 
value  of  the  complex  number. 
We  frequently  indicate  the  modu- 
lus or  absolute  value  of  any  com- 
plex number  a  by  placing  a  vertical 
line  before  and  after  the  number,  thus  \  a\,  read  "the  absolute  value 
of  a." 

Other  geometric  interpretations  of  complex  numbers  are  possible. 
We  shall  have  occasion  later  to  point  out,  for  example,  how  complex 
numbers  may  be  represented  by  points  on  a  sphere  by  showing  that 
there  exists  a  one-to-one  correspondence  between  the  points  of  the 
complex  plane  and  those  on  the  surface  of  a  sphere. 

From  what  has  already  been  said,  it  will  be  seen  that  complex 
numbers  are  directed  numbers,  that  is,  numbers  that  have  both 
magnitude  and  direction.  Consequently,  we  may  when  convenient 
think  of  the  complex  number  a  + 16  as  fepresented  by  the  plane 
vector  joining  the  corresponding  point  of  the  complex  plane  with 
origin.  Such  physical  magnitudes  as  force,  velocity,  acceleration, 
electric  intensity,  etc.,  have  direction  as  well  as  numerical  value  and 
may  be  represented  therefore  by  complex  numbers,  provided  their 
directions  are  confined  to  a  plane.     The  factor  i  rotates  the  given 

number  through  an  angle  ^ .    Thus  ia,  as  we  have  seen,  indicates  that 

a  distance  a  is  to  be  laid  off  on  a  line  perpendicular  to  the  axis  of 
reals.     In  the  complex  number 

a.  =  p(cos  6  -{-  i  sin  6), 
the  magnitude  of  the  number  is  p,  while  the  direction  in  which  this 


8  REAL  AND  COMPLEX  NUMBERS  IChap.  I. 

magnitude  is  measured  is  determined  by  the  factor  in  the  paren- 
thesis. 

6.  Comparison  of  complex  nimibers.  The  question  very  natu- 
rally arises  as  to  how  two  complex  numbers  may  be  compared  with 
each  other.     Given  the  two  numbers 

a  =  a  -[■  iby        ^  =  c  -\-  id. 

We  say  that  a  =  /3,  when  we  have  the  relations 

a  =  c,        b  =  d. 

Expressed  in  terms  of  polar  coordinates,  equality  involves  the  con- 
dition that  the  two  numbers  shall  have  equal  moduli  and  shall  have 
amplitudes  that  are  either  equal  or  differ  by  some  multiple  of  2  w.  It 
will  be  observed  that  the  two  equal  numbers  a  and  /3  are  represented 
by  the  same  point  in  the  complex  plane. 

Since  the  moduli  of  complex  numbers  are  real,  their  magnitudes 
may  be  compared  one  with  another  in  the  same  manner  as  any  other 
real  numbers.  Thus  of  two  complex  numbers  a  and  /3,  it  is  possible  to 
say  that  the  modulus  of  a  is  greater  than  or  less  than  the  modulus 
of  /3;  that  is,  we  may  write 

l«|5  1/3  |. 

7.  Addition  and  subtraction  of  complex  numbers.  We  define 
the  sum  of  two  complex  numbers  a  -{-  ib  and  c  +  id  as  the  complex 
number  {a  -\-  c)  -\-  i  {b  -\-  d),  obtained  by  adding  the  real  parts  and 
the  imaginary  parts  separately. 

It  is  not  to  be  assumed  without  proof  that  the  laws  of  addition 
enumerated  for  real  numbers  in  Art.  3  hold  for  complex  numbers. 
It  may  be  easily  shown,  however,  that  such  is  the  case.  For  this 
purpose,  suppose  we  have  given  any  three  complex  numbers 

a  =  a  +  t6,        /3  =  c  +  id,        y  =  e-{-if. 
That  the  commutative  law  holds  is  shown  as  follows :    We  have 

a-\-^  =  (a-\-c)-\-iib  +  d) 

=  {c-{-a)-{-i{d-h  b)  (Art.  3) 

=  /3  +  a. 

To  show  that  the  associative  law  likewise  holds  we  proceed  as 
follows: 

a  +  (^  +  7)  =  [a  +  (c  +  e)]  +  i  [b  +  (d  +/)] 

=  [(a  +  c)  -f-  e]  +  i  [(6  +  d)  -f-/]       (Art.  3) 
=  (a  +  ^)  +  7- 


Art.  7.] 


ADDITION  AND  SUBTRACTION 


9 


^x 


Fig.  2. 


The  addition  of  complex  numbers  can  be  easily  performed  geo- 
metrically. In  Fig.  2,  let  a  =  a  +  ih  and  j8  =  c  +  id  be  represented 
by  the  points  R  and  S,  respectively.  Complete  the  parallelogram 
having  OR  and  OS  as  two  sides. 
The  point  P  then  represents  the 
sum  a  +  jS;  for,  drawing  through 
R  a  parallel  to  OX  and  dropping 
from  P  the  perpendicular  PM, 
we  have  from  the  equality  of  the 
two  triangles. -RlfP  and  ONS 
RM  =  c,        MP  =  d, 

and  consequently  the  coordinates 
of  P  are  (a  -\-  c,  b  -\-  d).  Hence, 
the  point  P  represents  the  complex 
number 

(a  +  c)  +  i  (6  +  rf)  =  a  +  ^/ 

The  result  of  geometric  addition  may  be  conveniently  obtained 
also  as  follows:  To  add  /3  to  a,  draw  from  R  (Fig.  2)  a  line  parallel 
to  OS,  extending  in  the  same  direction  and  equal  to  it  in  length. 
The  terminal  point  P  of  the  line  thus  drawn  represents  the  number 
a  -\-  ^.  To  add  several  numbers  in  succession,  all  that  is  necessary- 
is  to  draw  from  the  point  P  representing  the  sum  of  the  first  two 
numbers  a  line  parallel  to  the  line  on  which  the  modulus  of  the  third 
point  is  measured,  and  upon  the  line  thus  drawn  to  lay  off  from  P 
a  segment  equal  to  OP  and  extending  in  the  same  direction.  The 
terminal  point  of  this  line  represents  the  sum  of  the  first  three  num- 
bers. To  this  result  a  fourth  number  may  be  added  in  the  same 
way,  etc. 

An  important  relation  between  the  absolute  values  of  two  complex 
numbers  is  suggested  by  the  geometric  considerations  already  in- 
troduced.    This  relation  may  be  stated  as  follows: 

Theorem  I.  Given  two  complex  numbers  a  and  j3;  we  have  the 
follovnng  relation  between  their  moduli,  namely, 

1  |al-  l^ll^  |a-hi3l  ^  lal  +  I/Sl. 

From  elementary  geometry,  it  is  known  that  any  side  of  a  triangle 
is  greater  than  the  difference  of  the  other  two  sides  and  less  than  the 
sum  of  those  sides.     Referring  now  to  Fig.  2,  we  have 

OR  -  RP  =  OP  =  OR  +  RP.  (1) 


10 


REAL  AND  COMPLEX  NUMBERS 


[Chap.  I. 


But  we  have 

OR=\al 


RP  =  OS=\fi\,  OP=\a  +  P\. 


(2) 


Hence,  substituting  these  values  in  (1),  we  have  the  result  given  in 
the  theorem.     An  equality  sign  is  to  be  used  only  when  the  points 
representing  a  and  /3  lie  on  a  straight  line  passing  through  the  origin. 
We  may  state  also  the  following  theorem. 

Theorem  II.     For  a  given  complex  number  a  =  a  -\-  ib,  we  have 

\a\+\b\^V2\a-\-ib\.  (!) 

The  proof  of  this  theorem,  like  that  for  Theorem  I,  follows  at  once 
from  geometrical  considerations.  As  the  point  a  moves  around  the 
circle  (Fig.  3)  |  a  |  and  |  b  \  change  values.  Let  us  find  the  maxi- 
mum value  of  I  o  I  +  I  6  I,  subject 
to  the  condition  that 

|a|2+16|2=  |a  +  i6|2  =  r2,     (2) 

where  r  is  the  radius  of  the  given 
circle.  By  the  ordinary  methods 
for  computing  a  maximum,  we  find 
that  the  function  in  question  has 
a  maximum  if 

a  -\-  ib\ 


a    =    b    = 


Fig.  3. 


V2 


Consequently,  the  maximum  value  of  |  a  |  +  |  6  |  is  V^  \a  -\-  ib  \, 
from  which  the  relation  (1)  must  hold  for  the  various  values  of  a  upon 
the  circle. 

We  define  the  difference  a  —  /3  of  two  complex  numbers,  a  =  a-\-ib, 
/3  =  c  +  id,  as  the  complex  number  (a  —  c)  -\-  i  (b  —  d).  Here,  as 
in  addition,  the  laws  of  operation  with  real  numbers  hold  for  com- 
plex numbers  and  follow  as  a  consequence  of  the  laws  of  addition 
and  the  definition  of  subtraction.  The  reader  can  easily  verify  this 
statement. 

Let  a  and  /3  be  represented  by  R  and  *S,  respectively  (Fig.  4). 
To  subtract  jS  from  a  geometrically,  construct  the  parallelogram 
having  OR  as  a  diagonal  and  OS  as  one  side.  The  point  P  repre- 
sents a  —  /3,  since  the  sum  of  h;  and  the  complex  number  repre- 
sented by  this  point  is  )8.  The  same  result  is  obtained  by  drawing 
the  line  RP  parallel  to  SO  and  equal  to  it  in  length. 


Art.  8. 


MULTIPLICATION 


11 


r.. 

s/3 

/C^^^^-          ^ 

/  .       *       ^^ 

/  ^y^/ 

(^                                                                i              /                                                           s.      -T»- 

0 

8.   Multiplication  of  complex  numbers.    We  may  define  the  prod- 
uct of  two  complex  numbers  a,  /3  by  the  relation 
a^  =  {a-\-  lb)  (a'  +  ib') 

=  {aa'  -  bb')  +  i{ab'  +  fea'). 

If  the  given  numbers  are  written  in 
the  form 

a  =  pi(cos  01  +  i  sin  ^i), 
j8  =  p2(cos  02  +  i  sin  ^2), 

then  the  product  a/3  becomes 

a/3  =  pi(cos  01  +  I  sin  di)  •  p2(cos  02 
+  i  sin  ^2) 
=  PiP2[(cos0iCos02  — sin0isin02) 
+t(sin0icos02+cos0isin02)] 
=  piP2[cos  (01  +  ^2)  +  i  sin  (^i 

+  ^2)].  Fig.  4. 

This  relation  gives  us  the  rule  for  multiplication,  which  may  be 
stated  in  words  as  follows: 

The  product  of  two  complex  numbers  is  a  complex  number  whose 
modulus  is  the  product  of  the  moduli  of  the  two  given  numbers  and  whose 
amplitude  is  the  sum  of  the  given  amplitudes. 

The  associative,  commutative, 
distributive,  and  factor  laws  for 
multiplication  hold  for  complex 
numbers  as  for  real  numbers.  The 
commutative  law  for  example  can 
be  established  as  follows: 

Given  as  before  a,  /3,  we  have 
a/3  =  pip2[cos  (01  +  02)  +  i  sin(0i+  ^2)] 
=  p2pi[cos  (02  +  0i)  +  i  sin (02+  0i)] 
=  /3a.  (Art.  3) 

In  a  similar  manner  the  associa- 
tive, distributive,  and  factor  laws 
^^-  ^-  may  be  established. 

From  the  definition  of  a  product,  we  are  able  easily  to  give  a  geo- 
metric interpretation  of  multiplication.  Represent  the  two  complex 
numbers  a  and  /3  by  the  points  P  and  Q,  respectively.  Thus  far  we 
have  been  able  to  carry  out  the  geometric  operations  introduced 
without  reference  to  the  magnitude  of  the  unit.     We  must  now 


12  REAL  AND  COMPLEX  NUMBERS  [Chap.  I. 

make  use  of  a  unit  length.  Lay  off  on  OX  the  distance  OA,  which 
we  take  as  the  unit  of  length.  Connect  the  point  P  with  A.  On 
OQ  as  a  base  construct  the  triangle  OQR  similar  to  the  triangle  OAP. 
We  have  then  from  the  figure 

OR:OP::OQ:l; 
that  is, 

OR  :  pi : :  p2 : 1, 

or  P1P2  =  OR. 

Furthermore,  we  have  by  construction 

ZQOR  =  ZAOP; 
hence 

ZAOR  =  6i-\-d2. 

Consequently,  from  the  definition  of  multiplication  it  follows  that 
the  point  R  represents  geometrically  the  required  product;  for,  it 
has  the  modulus  pip2  and  the  ampUtude  di  +  62. 

The  rule  of  multiplication  may  be  extended  to  the  product  of  any 
finite  number  of  complex  numbers.     Suppose,  for  example,  we  have 


<xi  =  pi(cos  di  +  i  sin  di), 
0C2  =  P2(cos02  +  ismd2)f 


an  =  Pn(C0S  dn  +  i  slu  On). 

We  obtain  as  the  continued  product  of  these  numbers 

aiui  .  .  .  an  =  piP2  .  .  .  pn[cos(ei+  •  •  •  +0„)+isin(5i+  •  •  •  +  9n)] 

When  pi  =  P2  =   •  •  •  =  Pn  =  1,  we  have  an  important  theorem, 
known  as  De  Moivre's  theorem,  namely: 

(cos  61  -\-  i  sin  di)  •  (cos  62  +  i  sin  6^  .  .  .  (cos  dn-\-  i  sin  0„) 
=  cos  (di  +  02  +  •  •  •  +  Qn)  +  i  sin  (5i  +  02  +  •  •  •  0n). 

If  we  also  put  61=62=  •  •  •   =  0n  =  0,  we  have  the  form  of  the 
theorem  most  frequently  used,  namely: 

{cos  6  -\-i  sin  6Y  =  cosriB  -\-  i  sin  n6. 

This  theorem  gives  us  a  method  of  raising  any  complex  number  to 
an  integral  power;  for,  we  have 

[p(cos  6  -\-  isia  0)]"  =  p"(cos  n6  -\-  i  sin  nd). 


Art.  8. 


MULTIPLICATION 


13 


The  power  of  a  complex  number  may  be  found  geometrically  as 
follows:  Let  Pi  represent  the  number 

a  =  p(cos  6  -\-  i  sin  6). 

On  OPi  construct  the  triangle  OP1P2  similar  to  OAPi]  then  P2  rep- 
resents the  second  power  of  the 
given  complex  number.  On  OP2 
as  a  base  construct  another  tri- 
angle OP2P3  similar  to  OAPi  or 
to  OP1P2.  The  point  P3  of  this 
triangle  represents  the  cube  of  the 
given  number.  Continuing  in 
this  way,  we  may  raise  the  given 
number  to  any  required  integral 
power. 

It  may  be  shown  that  De 
Moivre's  theorem  holds  likewise 
for  negative  and  fractional  pow- 
ers.* As  an  iHustration,  let  us 
consider  the  n^^  roots  of  a  complex  number  a.     We  have  then 

a"  =  p"  COS  -  +  I  sm  - ), 
\      n  nj 

1 

where  p"  is  always  to  be  considered  positive  and  real.     We  obtain 

the  remaining  n^^  roots  of  a  by  replacing  6  by  d  -{-  2  kT,  k  =  1,  2,  3, 


Fig.  6. 


.,  n  —  1.     Thus  the  n  roots  are 


>"(( 


p"l  cos 


)"  cos  - 
V       n 


+  isin-)> 
n 


ism 


e^-2ir 


)' 


If       9-\-2(n-l)ir  ,    .   .    0  +  2  (n  -  1)  ttN 

p"i  cos ^ h  I  sm ^ —  • 

\  n  n  / 

That  each  of  these  numbers  is  an  n*^  root  of  a  is  seen  at  once  from 
the  fact  that  its  n*^  power  gives  the  number 

a  =  p(cos  d  -\-  i  sin  6). 


*  See  Hobson,  Plane  Trigonometry,  2d  Ed.  Arts.  181  and  186. 


14  REAL  AND  COMPLEX  NUMBERS  [Chap.  I. 

If  ^  be  the  chief  amplitude  of  a,  that  is  if  —  tt  <  0  =  tt,  then 
y      e  ,   .  .    6' 


p"  cos  -  +  1  sin  - 
\       n  nj 


is  called  the  chief  or  principal  value  of  the  root. 

For  example,  consider  the  positive  real  number  a  =  a.     The  two 
square  roots  of  a  are 

Va(cos  0  +  i  sin  0),        Va(cos  x  +  i  sin  tt). 

The  principal  value  of  a^  is 

Va(cos  0  +  i  sin  0)  =  Va; 
for,  in  this  case 

e  =  71'  amp  a"  =  2-0  =  0 

falls  in  the  interval  —K<d=ir. 

If  we  consider  the  number  a  =  —  a,  we  have  as  the  two  square 
roots 

Vol  cos  X  +  i  sin  ^j,  Va(  cos -^  +  ^  sin -^  j> 

and  the  principal  value  of  a^  is 


'Vai  cos  ^  +  1  sin  X  j  —  *  Va. 


As  a  further  illustration  of  the  use  of  De  Moivre's  theorem,  let  us 
consider  the  n  roots  of  unity.     Here  6  =  0,  and  we  have  as  the  roots 

cos  0  -\-  i  sin  0, 

27r   ,    .    .    27r 

cos h  i  sm  —  > 

n  n 

47r   ,    .   .    47r 

cos h  *  sm  — , 

n  n 


2  (n  -  1)  X  ,    .  .    2  (n  -  1)  IT 
cos— ^^ h  ism — 


If  we  denote  the  second  of  these  roots  by  w,  then  the  n  roots  may  be 
written 

1,   O),   co^,   .   .   .   ,   w"~^. 


Art.  8.] 

Since  we  have 


MULTIPLICATION 


cos 


d-\-2k-n- 


isin- 


6-^2  kir 


=  (cos-+  1  sin- 
\      n  n 


2k7r  ,    .   .    2kTr 

cos h  t  sin 

n  n 


we  may  write  the  n  roots  of  any  complex  number  a  in  the  form 


03a' ,  aj^a" 


I 


15 


(1) 


(2) 


where  a"  denotes  one  of  the  n  roots  of  a,  for  example  the  principal 
value  of  the  root;  that  is  to  say,  the  n  roots  of  any  complex  number 
are  the  products  of  some  one  value 
of  the  root  into  the  n  roots  of 
unity. 

In  certain  cases  the  roots  of  a 
complex  number  may  be  deter- 
mined graphically.  It  is  not  pos- 
sible, however,  to  do  this  in  all 
cases;  for,  it  is  not  always  possi- 
ble to  make  the  construction  by 
means  of  the  ruler  and  compasses. 
Let  us  consider  the  fourth  roots  of 
the  number  a,  represented  by  the 
point  P  (Fig.  7).  Denote  the  mod- 
ulus of  a  by  p  and  its  chief  amplitude  by  d. 
the  root  is  then  given  by 


>X 


The  principal  value  of 


=  p^y 


cos  -  -f- 1  sm  J 


e 


To  determine  this  root,  it  is  necessary  to  construct  an  angle  j  and 

d 


We  can  construct  the  angle  ^  by 


to  lay  off  a  distance  equal  to  p* 

dividing  twice  in  succession  the  angle  d  by  the  methods  of  elementary 
geometry.  We  can  find  the  line  segment  that  represents  p*  by  con- 
structing the  mean  proportional  between  OP  and  1  and  then  con- 
structing the  mean  proportional  between  that  result  and  1.  In  this 
manner  the  point  Q  is  determined  representing  the  principal  value 
of  the  fourth  root  of  a.  That  Q  does  represent  a  fourth  root  of  a, 
may  be  shown  by  constructing,  as  indicated  in  Fig.  6,  the  fourth 


16  REAL  AND  COMPLEX   NUMBERS  [Chap.  I. 

power  of  the  number  represented  by  Q.  There  are  three  other 
fourth  roots  of  a.    From  (2)  they  are  seen  to  be 

a*(i),  a'oj*,  Q!*&;', 

where  <a  denotes  a  fourth  root  of  unity.  To  multiply  by  w,  ur,  or  w' 
is  to  rotate  the  line  OQ  in  a  counter-clockwise  direction  through  an 
angle  of  90°,  180°,  270°,  respectively,  about  0  as  a  center.  To  find 
geometrically  the  four  points  representing  the  fourth  roots  of  a  is 
equivalent  to  constructing  a  regular  inscribed  polygon  of  four  sides 
in  a  circle  having  OQ  as  a  radius  and  Q  as  one  of  the  vertices.  Each 
vertex  of  this  polygon  is  a  fourth  root  of  a,  as  may  be  verified  by 
constructing  its  fourth  power. 

To  determine  graphically  all  of  the  n^  roots  of  any  complex  num- 
ber involves  the  division  of  the  chief  amplitude  into  n  equal  parts, 
the  laying  off  of  a  distance  equal  to  the  n^^  root  of  the  given  modu- 
lus, and  the  inscribing  of  a  regular  polygon  in  a  circle.  For  the 
special  case  of  the  n*'^  roots  of  unity,  the  problem  reduces  to  the 
construction  of  a  regular  polygon  inscribed  in  a  circle  of  unit  radius; 
for,  the  modulus  is  1  and  in  this  case  the  chief  amplitude  is  zero.  As 
has  already  been  pointed  out  this  construction  is  not  always  possible 
by  means  of  a  ruler  and  compasses.  The  construction  is  however 
possible  if  and  only  if  we  have  *  n  =  2'  pip2  .  .  .  ,  where  I  is  an  in- 
teger and  pi,  jh  ■  ■  •  are  distinct  prime  numbers  of  the  form  2^'  +  1. 
For  example,  it  is  possible  if 

n  =  3,  4,  5,  6,  8,  10,  12,  15,  16,  17,  20,  24,  .".  . 

and  impossible  if 

n  =  7,  9,  11,  13,  14,  18,  19,  21,  22,  23,  25,  ...  . 

9.  Division  of  complex  numbers.     Given  two  complex  numbers 

a  =  a  -\-  ib,         /3  =  a'  +  ib'. 

Since  division  is  the  inverse  operation  of  multiplication,  we  must  so 
define  the  quotient  of  a  by  jS  that  the  result  multiplied  by  /3  gives  a. 
In  accordance  with  this  relation,  we  define  the  quotient  of  a  divided 
by  /3  by  means  of  the  identity 

a  _  g  +  i&   _  oa'  +  66'       .  a'b  —  6b' 
^  ~  o'  +  i6'  ~  a'2  +  6'2  "^  *  a'2  +  V  ' 

*  See  Monographs  on  Modem  Mathenmlica,  edited  by  J.  W.  A.  Young,  p.  379. 


Art.  9.]  DIVISION  17 

Writing  a,  )3  in  the  form 

a.  =  pi(cos  di  -\- 1  sin  di),        /3  =  p2(cos  ^2  +  *  sin  ^2), 
the  quotient  of  a  by  /3  may  be  written  as  follows: 

^  =  ^  [cos  (01  -  02)  +  i  sin  {6,  -  d^)]. 

P  P2 

This  form  of  the  definition  may  be  expressed  in  words  as  follows: 

The  quotient  of  one  complex  number  by  another  is  a  complex  number 
whose  amplitude  is  the  amplitude  of  the  dividend  minus  that  of  the 
divisor,  and  whose  modulus  is  the  modulus  of  the  dividend  divided  by 
that  of  the  divisor. 

We  have  already  pointed  out  in  another  connection  that  the 
amplitude  of  a  complex  number  is  multiple-valued.  This,  however, 
does  not  affect  the  quotient;  for,  an  increase  of  the  amplitude  of 
the  dividend  or  the  divisor  by  a  multiple  of  27r  increases  or  de- 
creases likewise  the  amplitude  of  the  quotient  by  the  same  multiple 
of  2  TT  and  hence  the  result  remains  unchanged. 

From  the  definition  of  division,  we  have  for  the  reciprocal  of  a 
complex  number  a 

1  _  (cos  0  +  i  sin  0) 
a      p(cos  0  +  *  sin  6) 

=  i[cos(-0)  4-i;sin(-«)] 
P 

=  -  (cos 5  —  isind). 
p 

We  may  perform  geometrically  the  operation  of  division  as  follows: 
Let  P,  Q,  represent  the  two  complex  numbers 

a  =  pi(cos  01  -{-  i  sin  di) 
and 

/3  =  p2(cos  02 -\-  i  sin  ^2), 

respectively.  Draw  the  line  OM  (Fig.  8)  making  the  angle  —62  with 
OP.  Construct  on  OP  the  triangle  ORP  similar  to  OQA,  when  OA  =  1. 
The  point  R  represents  the  quotient;  for.  it  has  the  amplitude  61  —  62 

and  its  modulus  is  — »  since  we  have  from  the  two  similar  triangles, 

P2 

ORP  and  OAQ, 

OR  :  1  :  :  pi  :  P2, 

and,  hence,  —  =  OR. 

P2 


18 


REAL  AND  COMPLEX  NUMBERS 


[Chap.  I. 


It  will  be  observed  that  -  has  no  significance  when  ps  =  0;   for, 


division  by  zero  is  meaningless. 

n 


The  general  laws  of  division  for 
real  numbers  hold  for  complex 
numbers  and  are  a  consequence 
of  the  definition  of  division  and 
the  laws  of  operation  governing 
multipUcation. 

We  have  now  defined  t"he  funda- 
mental operations  of  arithmetic 
with  reference  to  complex  num- 
bers. Moreover,  we  have  seen' 
that  the  general  laws  of  opera- 
tion in  the  arithmetic  of  real  num- 
bers may  be  extended  without 
modification  to  complex  numbers. 
We  are  now  in  a  position  to  in- 
troduce the  complex  variable  and 
functions  of  it.  Certain  funda- 
mental notions  concerning  the 
functions  to  be  considered  will  be  discussed  in  the  next  chapter. 


EXERCISES 


«/3 


1.  Express h  7*  in  the  form  A  +  iB,  where  a,  0,  y  are  given  complex 

7 
numbers. 

y  2.  Perform  graphically  the  operations  indicated  by  (a/3  —  7)  -^  7>  («^  ■=■  7)  —  1, 
where  a=  2  +  i  3,  0  =  1  +  i  2,  y  =  3  -  i  2. 

3.  Represent  graphically  the  square  roots  of  3  +  i  2,  —  2  +  i  3,  |  +  i  J. 

4.  Represent  geometrically  the  four  values  of  (1  +  V  — 1)S 

5.  Locate  the  points  representing  the  sixth  roots  of  1  and  of  —1. 

6.  Give  an  illustration  of  two  complex  numbers  the  sum  of  whose  moduli 
is  equal  to  the  modulus  of  their  sum;  also  two  complex  numbers  the  sum  of  whose 
moduli  is  greater  than  the  modulus  of  their  sum. 

7.  Under  what  conditions  do  the  relations 

|a  +  6I  =  |a|  +  IM, 
|a  +  6|  =  |a|-16|, 

hold  when  a,  b  are  real  numbers?  Under  what  conditions  do  they  hold  when  a, 
b  are  replaced  by  the  complex  numbers  a,  /3? 

■7  8.  Prove  the  associative  and  distributive  laws  for  multiplication  of  complex 
numbers. 

^9.   If  o  and  /8  are  complex  numbers,  where  a  ^0,  fi  ^0,  prove  that  afi  ^0. 


Art.  9.]  EXERCISES  19 

.;    10.   Interpret  geometrically  (a/S)"*  =  a^^;  cJ^a^  =  01"^+^. 

11.  A  boat  is  being  rowed  from  the  west  to  the  east  bank  of  a  stream  at  the 
rate  of  three  miles  per  hour.  At  the  same  time  it  is  being  carried  north  by  the 
current  at  the  rate  of  two  miles  per  hour.  Represent  the  velocity  by  a  point  in 
the  complex  plane.     What  represents  the  speed? 

12.  A  fly-wheel  two  feet  in  diameter  is  revolving  counter-clockwise  at  the 
rate  of  180  revolutions  per  minute.  Express  as  a  complex  number  of  the  form 
a  -{■  ih  the  velocity  of  a  point  on  the  rim  of  the  wheel  where  the  radius  vector 
through  that  point  makes  an  angle  <f>  with  a  fixed  initial  position. 


CHAPTER  II 
FUNDAMENTAL  DEFINITIONS  CONCERNING  FUNCTIONS 

10.  Constants,  variables.  We  shall  make  the  same  distinctions 
between  constants  and  variables  as  in  the  realm  of  real  variables. 
If  a  complex  number  assumes  but  a  single  value  in  any  discussion, 
it  is  called  a  constant.  The  numbers  thus  far  considered  serve  as 
illustrations.  If,  on  the  other  hand,  a  number  is  allowed  in  any 
discussion  to  assume  various  complex  values,  it  is  called  a  variable. 
A  complex  variable  z  may  be  written  in  the  form  x  +  iy,  where 
X  and  y  are  real  variables. 

We  shall  speak  of  a  connected  portion  of  the  complex  plane  as  a 
region  or  domain.  Any  point  Zo  is  said  to  be  an  inner  point  of  a 
region  if  it  can  be  made  the  center  of  a  circle  of  radius  different 
from  zero  such  that  all  points  within  this  circle  are  points  of  the 
region.  If  the  circle  can  be  taken  so  small  that  it  contains  no  points 
of  the  region,  then  2o  lies  exterior  to  the  region.  If  the  circle  contains 
both  points  of  the  region  and  points  exterior  to  it,  however  small 
the  radius  of  the  circle  be  taken,  then  z^  is  called  a  boundary  point 
of  the  region.  If  the  boundary  is  included  in  the  region,  then  it  is 
spoken  of  as  a  closed  region,  otherwise  it  is  called  an  open  region. 
Unless  the  contrary  is  stated,  the  term  region  will  be  used  to  desig- 
nate an  open  region,  that  is,  a  connected  portion  of  the  complex  plane 
consisting  only  of  inner  points.  A  given  region  may  be  finite  or  it 
may  be  infinite.  The  inner  points  of  an  infinite  region  may  all  he 
exterior  to  a  given  curve  in  which  case  the  curve  is  the  boundary  of 
the  region  if  its  points  are  boundary  points.  By  a  neighborhood  of  a 
given  point  zo,  we  shall  understand  a  small  region  about  20  having  20 
as  an  inner  point.  For  most  purposes  it  will  be  found  convenient  to 
choose  a  neighborhood  for  which 

I  2  -  2o  I  <  P, 

that  is  a  circle  of  radius  p  about  the  point  2o-  In  referring  to  the 
points  of  a  neighborhood  of  zo  exclusive  of  the  point  20  itself,  we  shall 
speak  of  the  region  as  a  deleted  neighborhood  of  2o. 

20 


Art.  11.]  CLASSIFICATION  OF  FUNCTIONS  21 

11.  Definition  and  classification  of  functions.  The  definition  of 
a  function  has  undergone  radical  changes  since  it  was  first  intro- 
duced in  connection  with  real  variables.  Leibnitz,  for  example, 
associated  the  term  with  any  expression  standing  for  certain  lengths 
connected  with  curves,  such  as  tangents,  radii  of  curvature,  normals, 
etc.  Euler,  who  wrote  the  first  treatise  on  the  theory  of  functions, 
defined  a  function  as  an  analytic  expression  in  which  one  or  more 
variables  appear.  We  must  set  aside  such  a  definition  because 
there  are  numerous  illustrations  of  related  and  interdependent  vari- 
ables, both  in  pure  mathematics  and  in  theoretical  physics,  where 
as  yet  no  analytic  expression  has  been  found.  These  early  defini- 
tions of  a  function  also  assumed  that  a  continuous  function  can 
always  be  represented  geometrically  by  a  continuous  curve  having  a 
definite  tangent  at  each  point.  This  condition  involves  the  require- 
ment that  every  continuous  function  shall  have  a  definite  derivative 
for  each  value  of  the  variable.  Subsequent  researches  show  that  this 
is  an  erroneous  assumption,  and  that  there  exist  functions  defined  by 
analytic  expressions  that  are  continuous  throughout  an  interval  and 
that  do  not  possess  a  derivative  at  any  point  of  that  interval.  The 
study  of  Fourier's  theory  of  heat  led  Dirichlet  in  1837  to  formulate 
the  following  definition  of  a  function  of  a  real  variable,  which  is  the 
one  commonly  accepted  by  mathematicians  at  the  present  time:  * 

If  for  each  value  of  a  variable  x,  there  is  determined  a  definite  value 
or  set  of  values  of  another  variable  y,  then  y  is  called  a  function  of  x  for 
those  values  of  x. 

This  definition  does  not  necessitate  the  existence  of  any  analytic 
relation  between  y  and  x.  It  is  to  be  observed  that  it  is  not  neces- 
sary that  x  should  have  every  value  in  an  interval;  it  may  take,  for 
example,  only  a  set  of  values.  What  is  essential  in  the  definition  is 
that  for  every  value  that  x  does  take,  there  is  thereby  determined  a 
definite  value  or  definite  values  of  y.  An  important  step  in  the  evo-< 
lution  of  the  idea  of  a  function  was  made  when  Cauchy  gave  to  thQ 
variable  complex  values,  and  extended  the  notion  of  a  definite  inte- 
gral by  letting  the  variable  pass  from  the  one  limit  of  integration 
to  the  other  through  a  succession  of  complex  values  along  arbitrary 
paths.  The  work  of  Cauchy  and  the  subsequent  work  of  Riemann 
and  Weierstrass  laid  the  foundation  for  the  development  of  the  gen- 
eral theory  of  functions. 

*  See  Enqjdopedie  des  Sciences  Math.,  Tome  II,  Vol.  I,  Fasc.  1,  p.  13. 


22  DEFINITIONS  CONCERNING  FUNCTIONS         [Chap.  II. 

We  shall  understand  the  complex  variable  w  to  be  a  function  of  the 
complex  variable  z  in  a  given  open  or  closed  region  S  if  for  each  value 
of  z  in  this  region  w  has  a  definite  value  or  set  of  values.  Here,  as  in 
the  case  of  functions  of  a  real  variable,  the  function  may  be  defined 
also  with  respect  to  a  set  of  values  rather  than  for  all  values  of  z  of 
a  given  region.  Unless  otherwise  stated,  however,  it  will  be  under- 
stood that  z  takes  all  values  of  a  given  region.  Then,  if  ty  is  a  func- 
tion of  z,  we  may  write 

w  =  u{x,y)  -\-  i  v{x,  y)  =  f(z), 

where  u,  v  are  real  functions  of  the  two  real  variables  x  and  y.  If  w 
has  but  one  value  for  each  value  of  z,  w  is  said  to  be  single-valued; 
if  it  takes  two  or  more  values  for  some  or  all  of  the  values  of  z,  then 
w  is  called  a  multiple-valued  function  of  z. 

A  possible  criticism  of  Dirichlet's  definition  is  that  it  is  not  suffi- 
ciently restrictive.  Without  introducing  some  additional  proper- 
ties, such  as  continuity,  differentiability,  etc.,  it  is  impossible  upon 
the  basis  of  this  definition  of  a  function  to  build  a  theory  of  analysis 
permitting  the  operations  and  developments  similar  to  those  of  the 
calculus  of  real  variables.  In  later  articles  we  shall  discuss  certain 
characteristic  properties  that  the  functions  to  be  considered  in  this 
volume  must  possess.  Before  doing  so  we  shall  define  certain  general 
classes  of  functions  and  discuss  some  of  the  fundamental  conceptions 
that  are  of  importance  in  the  consideration  of  their  properties. 

One  of  the  important  classifications  of  functions  is  their  division 
into  rational  functions  and  irrational  functions.  By  a  rational  in- 
tegral function  or  polynomial  is  understood  a  function  of  the  form 

an2"  +  a„-i2"~^  +  •  •  •  +  ao, 

where  oo,  ai,  .  .  .  ,  an  are  constants  and  n  is  a  positive  integer.  If 
a„  7^  0,  the  function  is  said  to  be  of  the  n*  degree. 

A  rational  fractional  function  is  the  quotient  of  two  rational  inte- 
gral functions  having  no  conmion  factor  and  hence  it  is  of  the  form 

UnZ"   +  «n-l2"~^  +     •    •    •     +  Cto^ 
PmZ"'-[- ^m-lZ"'-'  -\-     •    •    •     +/3o' 

where  m  and  n  may  be  equal  or  different  positive  integers.  If  a„  9^  0, 
/3m  ^  0,  and  m  =  n,  then  this  common  value  is  called  the  degree  of 
the  function;  if  m  ?^  n,  then  the  larger  of  the  two  is  called  the 
degree. 


Art.  12.]  LIMITS  23 

All  functions  which  are  not  rational  are  classified  as  irrational 
functions. 

Another  important  classification  of  functions  is  into  algebraic  and 
transcendental.  We  say  that  w  is  an  algebraic  function  of  z  when 
w  and  z  are  related  by  an  irreducible  equation  of  the  form 

/o(0)ttf»+/i(0)w;"-^+/2(0)w"-2+  •  •  .  +/„(2)  =0, 

where/o(2),/i(z),/2(2),  .  .  .  ,/„ (2)  are  rational  integral  functions  of  2. 
We  say  that  this  equation  is  irreducible  if  it  is  not  possible  to  write 
the  left-hand  member  as  the  product  of  two  polynomials,  neither  of 
which  is  a  constant.  It  will  be  seen  that  all  rational  functions,  for 
example,  are  algebraic  functions. 

All  functions  that  are  not  algebraic  are  called  transcendental 
functions.  Such  functions  include  the  trigonometric,  exponential, 
and  logarithmic  functions. 

12.  Limits.  From  the  study  of  elementary  mathematics,  and 
particularly  from  the  study  of  the  calculus,  the  student  is  familiar 
with  the  general  notion  and  properties  of  limits.  We  shall  recall 
some  of  the  fundamental  properties  by  way  of  emphasis  and  extend 
the  notion  of  a  limit  to  the  realm  of  complex  numbers. 

If  we  have  given,  for  example,  the  sequence  of  numbers 

it  is  at  once  seen  that  by  taking  n  sufiiciently  large  the  terms  of  the 
sequence  can  be  made  to  ultimately  differ  from  unity  by  as  little  as 
we  please.  We  express  this  fact  by  saying  that  the  sequence  has 
the  limit  1.     Likewise,  the  sequences 

111  J_ 

■■■>    4>    6>    •    •    •    J     2n'    •    •    * 

■'^J      4>     7>      •      •      •      >      „25      •      •      • 

have  the  limit  zero.  If  we  may  assign  at  pleasure  to  a  number 
values  which  are  numerically  as  small  as  we  may  choose,  then  the 
number  is  said  to  be  arbitrarily  small.  We  shall  usually  denote 
such  a  number  by  e.  We  may  now  define  the  limit  of  a  sequence 
more  exactly  as  follows: 

Suppose  we  have  given  an  infinite  sequence  of  real  numbers 

\an\  =  tti,  02,  as,  .  .  .  ,  a„,  .  .  .  . 


24  DEFINITIONS  CONCERNING  FUNCTIONS         [Chap.  II. 

If  there  exists  a  definite  number  a,  and,  corresponding  to  an  arbi- 
trarily small  positive  nmnber  e,  a  positive  integer  m  such  that  for  all 
values  of  n  >  w,  we  have 

I  ttn  -  a  I  <  e, 

then  a  is  called  the  limit  of  the  sequence,  and  we  write 

L  an  =  a. 

n=<» 

If  we  have  given  an  infinite  sequence  of  complex  numbers,  the 
moduli  of  these  complex  numbers  form  a  sequence  of  real  numbers, 
and  so  do  the  moduli  of  the  differences  between  these  complex  num- 
bers and  any  complex  constant.  We  say  that  a  sequence  of  complex 
numbers 

oci,  a2,  as,   ...  .  ,  ocn,  .  .  . 

has  the  limit  a  or  converges  to  the  limit  a,  if  the  moduli  of  the  differ- 
ences between  these  complex  numbers  and  a  form  a  sequence  having 
the  limit  zero;  that  is,  if  corresponding  to  an  arbitrarily  small  posi- 
tive number  c,  it  is  possible  to  find  a  positive  integer  m  such  that 
we  have 

I  a„  —  a  I  <  €,         n>  m.  (1) 

We  then  write 

L  an  =  a. 

n=oo 

Since  the  relation  given  by  (1)  holds  for  all  integral  values  of 
n  >  m,  it  likewise  holds  for  a  particular  set  of  values  of  w  >  m,  say 
for  the  even  values  of  w  >  m.  In  other  words,  any  subsequence 
selected  from  the  given  sequence  will  have  the  same  limiting  value 
as  the  given  sequence. 

The  foregoing  definition  of  the  limit  of  a  sequence  may  be  expressed 
in  terms  of  a  and  6,  where 

a  =  a  +  ib,         an  =  (in-\-  ibn) 

for,  we  have  the  following  theorem. 

Theorem  I.  The  necessary  and  sufficient  condition  thai  the  se- 
quence of  complex  numbers 

oci,  at,  as,  ...  ,  On,  ... 

converges  to  a  limit  a  =  a  -\-  ib  is  that 

L  On  =  a;        L  6„  =  6.  (2) 


Art.  12.]  LIMITS  25 

We  have 


whence 
and 


an  —  a  =  an  +  ibn  —  a  —  ib 

=  (ttn  -  a)  +  iih„  -  h), 

I  a„  —  a  I  =  I  a„  -  a  I  +  I  6„  -  &  I,  (3) 

I 

a„  -  a  I  =  I  a„  -  a  I  ,  |  6„  -  6  |  =  |  a„  -  a  ].  (4) 


The  condition  stated  in  the  theorem  is  necessary;  for,  if  the  given 
sequence  has  the  limit  a,  we  may  write 

I  a„  —  a  I  <  € 

for  n  sufl&ciently  great.     Hence,  from  (4)  we  have  also 

I  a„  —  a  I  <  e,          |  6„  -  6  |  <  e; 

that  is,  L  a„  =  a,        L  bn  =  b. 

n=oo  n=<x> 

The  given  condition  is  also  sufficient;  for,  if  the  two  limits  (2) 
exist,  we  thus  have  for  n  sufficiently  great. 

I  On  —  a  I  <  c,         I  6„  —  6  I  <  e, 

and  hence  from  (3)  it  follows  that 

|a„-al  <2€. 

Therefore,  the  given  sequence  has  the  limit  a  as  the  theorem  requires. 

Suppose  the  variable  z  takes  a  set  of  values  dense  at  a,  that  is,  a 
set  of  values  such  that  in  every  neighborhood  of  a,  however  small, 
there  are  an  infinite  number  of  points  representing  values  of  z.  We 
express  this  relation  between  z  and  a  by  saying  that  a  is  a  limiting 
point  of  the  variable  z.  Under  these  conditions  the  variable  z  may 
be  said  to  approach  its  limiting  value  a;  that  is,  it  may  so  vary  that 
I  2  —  a  I  decreases  indefinitely.  When  z  varies  in  this  manner,  we 
write z  =  a,  which  is  to  be  read  "as  z  approaches  a."  In  the  discus- 
sions to  follow  the  given  set  of  values  of  z  will  usually  include  every 
value  in  the  neighborhood  of  a,  and  unless  otherwise  stated  this  will 
be  understood  to  be  the  case. 

As  the  variable  z  takes  different  values,  any  single-valued  function 
of  z,  say  f{z),  has  by  definition  a  definite  value  for  each  value  z  is 
allowed  to  take.  The  values  of  f(z),  corresponding  to  the  values  of 
0  in  a  suitably  small  deleted  neighborhood  of  a  limiting  point  a,  may 
likewise  differ  from  some  number  A  by  amounts  whose  numerical 
values  are  less  than  an  arbitrarily  small  positive  number.    We  then 


• 


26  DEFINITIONS  CONCERNING  FUNCTIONS  [Chap.  IL 

speak  of  the  number  A  as  the  Umiting  value  of  the  function  f{z) 
corresponding  to  the  limit  a  of  z.  We  may  also  say  that  f{z) 
approaches  ^  as  2  approaches  a;  for,  under  the  conditions  just 
stated,  no  matter  how  z  may  approach  a,  j{z)  will  at  the  same  time 
approach  A.  We  may  now  formulate  the  definition  of  the  limit  of 
a  function  as  follows: 

If  a  is  a  limiting  point  of  z,  and  if  corresponding  to  an  arbitrarily 
^small  positive  number  e  there  exists  a  positive  numbei'^uch  that  for 
all  values  of  z  entering  into  the  discussion  for  which  |  z  —  a  |  <  5, 
with  the  possible  exception  oi  z  =  a,  we  have 

\f{z)-A\<e,  (5) 

then  f{z)  is  said  to  have  the  limiting  value  A  corresponding  to  the 
limit  a  of  z.  We  indicate  the  existence  and  the  value  of  this  limit 
by  writing 

L  f{z)  =  A.  (6) 

The  limit  of  a  function  does  not  depend  upon  the  value  of  the 
function  for  the  limiting  value  of  the  variable,  but  only  upon  the 
values  that  the  function  takes  in  the  deleted  neighborhood  of  such 
a  point.  Frequently  the  value  of  the  function  at  the  point  is  quite 
different  from  its  limiting  value.  So  far  as  the  mere  existence  of 
the  limit  is  concerned,  it  is  not  essential  that  the  function  be  defined 
for  the  limiting  value  of  the  variable.  The  general  laws  of  operation 
with  limits  as  developed  in  the  algebra  and  used  in  the  calculus  of 
real  variables  hold  equally  well  for  complex  variables,  as  they  are 
developed  without  reference  to  any  particular  domain  of  numbers. 

As  we  have  already  seen,  the  existence  of  the  limit  of  a  function 
involves  the  condition  that  the  same  limiting  value  of  f{z)  is  obtained 
whatever  be  the  set  of  values  through  which  z  is  permitted  to  ap- 
proach the  critical  value  a.  As  z  may  be  written  in  the  form  x  +  iy, 
where  x  and  y  are  independent  real  variables  and  a  is  a  number  of 
the  form  a  -\-  ib,  it  will  be  seen  at  once  that  the  limit  given  in  (6) 
is  related  to  the  double  simultaneous  limit  L  F(x,  y),  discussed  in 

x=a 
V=b 

connection  with  functions  of  two  jeal  variables,*  the  existence  of 
which  requires  that  the  same  limiting  value  be  obtained  by  all  pos- 
sible methods  of  approach  of  the  variable  point  (x,  y)  to  limiting 
position  (a,  h). 

*  See  Townsend  and  Goodenough,  First  Course  in  CoLcuLxts,  Arts.  101  and  102. 


Art.  12.1  LIMITS  27 

A  necessary  and  sufficient  condition  for  the  existence  of  the  limit 
(6)  may  be  stated  as  follows: 

Theorem  II.  Given  z  =  x  -\-  iy,  a  =  a  -]-  ib,  ^  =  A  -\-  iB;  the  nec- 
essary and  sufficient  conditions  that  J{z)  ^  u  -\-  iv  approaches  ^  as  z 
approaches  a  are  that 

L  u(x,  y)  =A,        L  v(x,  y)  =  B.  (7) 

1=0  •  i=a 

y=6  V=6 

To  prove  that  the  conditions  stated  in  the  theorem  are  necessary, 
we  have  given 

L  f(z)  =  fi,  (8) 

to  show  that  the  two  limits  (7)  exist.     From  (8),  we  have 

|/(z)-^|<6,         \z-a\<8. 
This  relation  may  be  written 

\u-hiv-  A  -iB\<€,  (9) 

for  values  of  (x,  y)  that  lie  within  the  circle  of  radius  8  about  the 
point  (a,  b);  that  is,  for  V{x  —  aY  -h  (y  —  by  <  5.  By  aid  of 
Theorem  II  of  Art.  7,  we  now  have  from  (9) 

\u  -  A  \  +  \v  -  B  \  ^  V2  \  (u  -  A)  -{-  i{v  -  B)  \  <  eV2. 
Hence,  it  follows  that 

\u-A\<eV2,     \v-B\<eV2,     V(x  -  a^  +  {y  -  bf  <  5. 
Expressed  in  the  form  of  Umits,  this  result  is 

L  u{x,  y)  =  A,        L  v{x,  y)  =  B. 

x=a  x=a 

y=b  y=b 

We  may  show  as  follows  that  the  given  conditions  are  also  suffi- 
cient. WeJiave  given  the  two  limits  (7)  to  show  the  existence  of 
the  hmit  (f^.     Expressed  as  inequahties,  the  limits  in  (7)  give 

I  u{x,  y)-A\<e,         V{x  -  a)'  +  (y  -  by  <  5,,  (10) 

I  vix,  y)  -  B\<e,         V(x  -  ay  -\- (y  -  by  <  8^.  (11) 
From  Theorem  I  of  Art.  7,  we  have 

\(u-A)-hi(v-B)\^\u-A\-\-\v-B\.  (12) 


28  DEFINITIONS  CONCERNING  FUNCTIONS  (Chap.  II. 

By  use  of  (10)  and  (11)  this  relation  becomes 

\u  +  w-A-iB\<2e,        V{x  -  a^  +  (y  -  by-  <  5',     (13) 

where  5'  is  the  smaller  of  the  two  numbers  5i,  52. 
Hence,  the  relation  (13)  may  be  written 

\f{z)-fi\<2e,  |2-al<5'. 

Expressing  this  result  in  terms  of  a  limit,  we  have 

L  f(z)  =  ^, 

as  the  theorem  requires. 

If  for  a  set  of  real  numbers  there  exists  a  definite  number  A  such 
that  the  numbers  of  the  set  never  exceed  A  but  are  dense  at  A,  then 
A  is  called  the  upper  limit  of  the  set.  For  example,  the  set  of  ele- 
ments constituting  the  sequence 

A  2>    A  3>    ■'^  i)    '    '    •    )    ^  n'    '    '    ' 

has  the  upper  limit  1.  In  this  particular  case  the  number  1  is  not 
an  element  of  the  set,  although  the  elements  approach  the  value  1. 
It  is  possible  that  the  upper  limit  of  a  set  shall  be  itself  an  element 
of  the  set.  When  this  is  the  case,  we  call  the  upper  limit  of  the  set 
the  maximum  value  of  the  set.  Thus,  in  the  sequence  just  given  1 
is  the  upper  Umit  of  the  set  but  not  the  maximum  value  of  the  set. 
Likewise,  if  none  of  the  elements  of  a  set  of  real  values  are  smaller 
than  a  definite  number  A  and  the  set  is  dense  at  A,  then  the  number 
A  is  called  the  lower  limit  of  the  set.  If  the  lower  limit  is  at  the 
same  time  an  element  of  the  set,  it  is  called  the  minimum  value  of 
the  set.    For  example,  elements  of  the  sequence 

■J    Oi    oi     oi  oJ. 

">     ^2}     ^'iy     ^47     •     •     •     >     •^„>     •     •     • 

form  a  set  having  the  lower  hmit  2.  It  does  not,  however,  have  a 
minimum  value  as  2  is  not  an  element  of  the  set.  In  other  words 
there  is  no  smallest  number  in  the  set. 

While  the  terms,  upper  limit,  lower  limit,  maximum,  minimum, 
are  defined  with  reference  to  a  set  of  real  values,  they  may  be  ex- 
tended without  modification  to  the  absolute  values  of  a  function 
f{z)  of  a  complex  variable. 

If  the  elements  of  a  given  sequence  all  lie  in  a  finite  interval,  the 
sequence  is  often  spoken  of  as  a  bounded  sequence.    Every  bounded 


Art.  12.]  LIMITS  29 

sequence  of  increasing  real  numbers  has  a  limit.*  If  the  sequence  is 
not  bounded,  but  its  elements  increase  without  limit,  we  say  that 
the  sequence  becomes  infinite  and  indicate  that  fact  frequently  by 
writing  the  limit  equal  to  +  oo .  The  sign  of  equaUty  in  this  con- 
nection is  not  to  be  confused  with  the  ordinary  use  of  that  symbol 
and  should  be  understood  as  a  brief  and  convenient  way  of  writing 
"  becomes  infinite  "  or  "  increases  without  limit."  If  the  elements 
of  a  sequence  decrease  without  limit,  we  place  the  limit  equal  to 
—  00  and  say  that  the  sequence  becomes  negatively  infinite. 

We  shall  have  occasion  frequently  to  consider  also  an  infinite  se- 
quence of  regions  of  the  complex  plane  so  related  that  each  is  smaller 
than  the  one  preceding  it  and  is  contained  in  it.  However  small  one 
of  the  regions  in  the  sequence  may  be,  it  contains  nevertheless  an  in- 
finite number  of  points.  It  is  of  importance  to  be  able  to  say  when 
the  limit  of  such  a  sequence  is  a  single  point.  This  result  will  be 
estabUshed  if  it  can  be  shown  that  every  set  of  points  that  can  be 
chosen  by  selecting  in  any  manner  whatever  a  point  within  or  on  the 
boundary  of  each  region  has  necessarily  the  same  limiting  point  as 
the  regions  are  taken  smaller  and  are  made  to  approach  zero  in  area. 
We  shall  consider  two  special  cases,  namely,  a  sequence  of  circles 
and  a  sequence  of  rectangles.  In  this  discussion,  we  shall  make  use 
of  a  well-known  theorem  f  of  the  theory  of  functions  of  real  variables 
which  states  that  if  there  is  given  a  definite  sequence  of  intervals 
defined  by 

In  =    (an,    hn),  71  =    1,    2,    3,    .    .    . 

such  that  the  interval  /„  lies  in  /„_i  and  L  [Length  In]  =0,  then  if 

n=oo 

P„  is  any  point  in  7„,  end  points  included,  every  such  set  \Pnl  of 
points  has  an  unique  limit  p,  and  we  say  that  the  sequence  of  inter- 
vals \In\  defines  the  point  p. 
We  shall  now  consider  the  following  theorem. 

Theorem  III.  Let  [Sni  =  Si,  S2,  .  .  .  ,  Sn,  .  .  .  he  an  infinite  se- 
quence of  circles  so  related  that  each  lies  in  the  preceding  one  and  more- 
over having  the  property  that 

L  Area  Sn  =  0; 

0  =  00 

then  the  sequence  \Sn\  defines  a  limiting  point  as  n  increases  indefinitely. 

*  See  Pierpont,  Theory  of  Functions  of  Real  Variables,  Vol.  I,  Arts.  101  and  109. 
t  Ibid.,  p.  82. 


30 


DEFINITIONS  CONCERNING  FUNCTIONS 


[Chap.  II. 


Let  each  circle  Sn  be  inclosed  in  a  square  by  drawing  tangent 
lines  parallel  to  the  X-axis  and  to  the   F-axis.    We  have  then  a 

sequence  of  intervals  on  each  axis 
(Fig.  9)  satisfying  the  conditions 
of  the  foregoing  theorem  quoted 
from  the  theory  of  functions  of 
real  variables.  The  sequence  of 
intervals 

(fll,  &l),  (02,  62),    .    .    .   ,   (fln,  hn),    .    .    . 

situated  on  the  X-axis  defines  in 
the  limit  a  definite  point  a.  Like- 
wise the  sequence  of  intervals 

(ci,  di),  (02,^2),  .  .  .  ,  {cn,dn),  .  .  . 

on  the  F-axis  defines  in  the  limit  a  point  b.  Drawing  through  a,  b 
lines  parallel  to  the  two  axes,  respectively,  the  intersection  deter- 
mines a  definite  point  a  +  ib  of  the  complex  plane,  which  is  the 
limit  of  every  sequence  of  pqints  fP„j  obtained  by  taking  a  point 
in  each  region  of  the  sequence  ISnl-  This  sequence  ]Sn\  therefore 
defines  the  point  a  +  ib.  It  will  be  noted  that  the  demonstration 
does  not  preclude  the  possibility 
that  any  or  all  of  the  circles  *Si, 
Si,  .  .  .  may  be  tangent  internally. 
In  certain  discussions,  it  is  more 
convenient  to  consider  a  sequence 
of  rectangles  obtained  as  follows. 
Suppose  we  have  given  the  rect- 
angle R,  Fig.  10.  Divide  this 
rectangle  into  four  equal  parts  by 
drawing  lines  parallel  to  the  two 
axes.  Select  one  of  these  four 
rectangles  and  divide  it  into  four  equal  parts  in  a  similar  manner. 
Consider  this  operation  as  repeated  an  indefinite  number  of  times. 
We  may  now  state  the  following  theorem  with  reference  to  the 
sequence  of  rectangles  obtained. 

Theorem  IV.  Given  an  infinite  sequence  of  rectangles  Ri,  Rz,  .  .  .  , 
Rn,  .  .  .  such  that  each  lies  in  the  preceding  one.  Let  the  limit  of  each 
dimension  of  Rn  approach  zero  as  n  increases  withoui  limit.  Then  the 
given  sequence  of  rectangles  defines  in  the  limit  a  definite  point  of  the  plane. 


n 


^X 


Fig.  10. 


Akt.  12.]  LIMITS  31 

The  method  of  proof  for  this  theorem  is  substantially  that  given 
for  Theorem  III. 

By  the  aid  of  the  foregoing  theorem,  we  readily  establish  the 
following  theorem. 

Theorem  V.  Every  infinite  set  of  points  in  a  finite  region  of  the 
complex  plane  has  at  least  one  limiting  point. 

Let  Ri  be  the  rectangle  inclosing  all  of  the  given  points.  Divide 
the  rectangle  Ri  into  four  equal  rectangles  by  drawing  lines  parallel 
to  the  sides  of  Ri.  At  least  one  of  these  smaller  rectangles,  say  R2, 
must  contain  an  infinite  number  of  points  of  the  given  set.  In  a 
similar  manner  divide  R2  into  four  equal  rectangles;  at  least  one  of 
these  rectangles,  say  -R3,  must  contain  an  infinite  number  of  the  points 
of  the  given  set.  Regard  this  method  of  division  as  continued  indefi- 
nitely.   We  get  a  sequence  of  rectangles 

Ri,  R2,  R3,  .  .  .  ,  Rji,  .  .  .  , 

satisfying  the  conditions  of  Theorem  IV  and  hence  having  a  limiting 
point,  say  Zq.  But  as  each  of  the  rectangles  Rn  contains  an  infinite 
number  of  the  points  of  the  given  set,  it  follows  that  Zo  is  a  limiting 
of  the  given  set.  There  may  or  may  not  be  other  limiting  points, 
but  in  every  case  there  must  be  at  least  one.  The  result  of  the 
theorem  can  also  be  expressed  by  saying  that  the  given  set  of  points 
is  dense  at  Zo- 

It  is  frequently  desirable  to  have  a  condition  which  is  necessary 
and  sufficient  for  the  existence  of  the  limit  of  a  sequence.  Such  a 
condition  is  given  by  the  following  theorem. 

Theorem  VI.     Given  the  sequence  of  complex  numbers 

ai,    a2,    as,    ...    ,    ««,    OCn+l,    •    •    ■    ,    OCn+k,    OCn+k+h    .    .    . 

the  necessary  and  sufficient  condition  that  this  sequence  has  a  limit  is 
that  corresponding  to  an  arbitrarily  small  positive  number  c  there  exists 
a  positive  integer  m  such  that 

I  an  -  an+k  I  <  e,         A;  =  1,  2,  3,  ...  ,         n  =  m. 

We  shall  first  show  that  the  given  condition  is  necessary.  For 
this  purpose  suppose  a  to  be  the  limit  of  the  sequence.  If  n  is  taken 
sufficiently  large,  say  n  =  m,  we  have  from  the  definition  of  a  limit 

\an  —  a\  <  2 ' 
I  a  —  Un+k  I  <  s ,       fc  =  1,  2,  3,  .  .  .  . 


32 


DEFINITIONS  CONCERNING  FUNCTIONS 


[Chap.  II. 


Combining  these  two  inequalities,  we  have 
I  an  —  ocn+k  I  <  €,  A;  =  1,  2,  3, 


n  =  m 


which  is  the  condition  set  forth  in  the  theorem. 

The  above  condition  is  also  sufficient;  that  is,  given  the  condition 

I  an  -  an+k  \  <  e,         fc  =  1,  2,  3,  .  .  .  ,         n  =  m,         (14) 

it  is  possible  to  show  that  the  sequence  has  a  limit.  By  virtue  of 
this  condition  we  can  find  among  the  a's  some  a„  such  that  all  of 
the  points  an+i,  a„+2,  ...  lie  within  some  arbitrarily  small  distance 

e  from  «„.     If  we  take  €  <  h,  then 
it  follows  from  (14)  that  all  of  the 
points  a„,  n  =  m,  lie  within  a  circle 
(Fig,  11)  having  am  as  center  and     ^ 
a  radius  |.     The  points  a„,  a  j'or-    t 
tiori,  lie  within  the  circle  Ci'lmVing  • 
am  as   center  and   a  radius  equal 
to    1.       Among   the    points   am+i, 
am+2,  •  •  •  ,  there  exists  some  one, 
say  ami,  such  that  we  have 

Fig.  11.  fc  =  1,  2,  3,  .  .  .   ; 

that  is,  all  points  a„,  n  =  mi,  lie  within  a  distance  of  \  from  a^i. 
These  values  then,  o  fortiori,  lie  within  the  circle  d  drawn  about 
«„,  as  a  center  with  a  radius  of  \.  Among  the  points  am,+i 
am,+2,  .  .  .  there  can  be  found  one,  say  nii,  such  that 

I  Otmt  —  amj+A;  |   <  |,  fc   =    1,    2,    3,     .    .    .    . 

All  of  these  remaining  points  «„,  n  =  W2,  a  fortiori,  lie  within  the 
circle  C3  about  the  point  owj  having  a  radius  of  \. 

It  will  be  observed  that  C2  lies  within  Ci,  and  Cz  within  C2.     Con- 
tinuing in  this  manner,  we  obtain  a  sequence  of  circles 

Cl,    C2,    C3,     .    .    .    ,    Cn,     •    •    • 

fulfilling  the  conditions  of  Theorem  III;  for,  each  circle  lies  within  the 
preceding  circles  and  their  radii,  which  are 


1    *    i  i 

-••>     2>     4>     •    •     •    >     2'>'    •    •    •    » 


J 


Art.  13.]  CONTINUITY  33 

respectively,  have  the  limiting  value  zero.  Hence,  the  sequence  of 
circles  defines  a  definite  limiting  point,  which  we  may  designate 
by  a.    The  number  a  is  then  the  limit  of  the  sequence  ai,  az,  as,  ...  . 

13.  Continuity.  A  function  of  a  complex  variable  is  said  to  be 
continuous  at  a  point  if  the  value  of  the  function  at  that  point  is 
equal  to  the  limit  of  the  values  assumed  by  the  function  in  every 
neighborhood  of  the  point.  There  are  three  things  involved  in  con- 
tinuity; first,  the  function  must  be  defined  at  the  point  in  question; 
second,  the  function  must  have  a  unique  limit  as  the  variable  ap- 
proaches the  critical  value;  and  third,  the  value  of  the  limit  must  be 
equal  to  the  value  of  the  function  at  the  point.  If  either  one  of  the 
two  latter  conditions  fails,  the  function  is  said  to  be  discontinuous. 
If  the  first  condition  is  not  satisfied,  then  we  can  not  discuss  the 
continuity  of  the  function  at  the  point  in  question;  for,  the  function 
does  not  exist  at  that  point. 

This  definition  does  not  differ  in  form  from  the  definition  given  in 
calculus  for  the  continuity  of  a  function  of  a  real  variable,  in  which 
we  say  that  a  function  f{x)  of  a  real  variable  x  is  continuous  at 
a;  =  a,  if 

Lf{x)^f{a).  (1) 

x=a 

This  definition  requires  that  the  same  limiting  value  /(a)  be  obtained 
by  every  possible  approach  to  the  point  x  =  a,  that  is,  from  either 
the  right  or  the  left  and  through  any  set  of  values  dense  at  the  point 
a  that  may  be  chosen  from  those  values  that  x  may  take  in  the 
neighborhood  of  a.  It  is  necessary  that  we  take  into  consideration 
all  such  values  of  x  in  the  neighborhood  of  re  =  a,  in  determining  the 
existence  or  non-existence  of  the  limit. 

In  a  similar  manner,  we  say  that  a  function  of  two  real  variables 
f{Xj  y)  is  continuous  at  the  point  (a,  h)  with  respect  to  the  two 
variables  taken  together  if  we  have 

L  fix,  y)=f(a,  6),  .  (2) 

.  1=0 

»=6 

which  involves  the  condition  that  the  same  limiting  value  is 
obtained  by  all  possible  methods  of  approach  to  the  point  (a,  6) 
and  furthermore,  that  this  limiting  value  is  equal  to  the  value  of  the 
function  at  that  point. 

The  definition  of  the  continuity  of  a  function  f{z)  of  a  complex 


34  DEFINITIONS  CONCERNING  FUNCTIONS  [Chap.  II. 

variable  may  be  briefly  expressed  by  saying  that  /(s)  is  continuous 
at  an  inner  point  z  =  a  oi  &  region  S,  if  we  have 

Lf(z)=f(a).  (3) 

More  is  involved  in  this  definition  than  in  the  corresponding  defini- 
tion for  continuity  of  a  function  of  a  single  real  variable.  The 
variable  x  has  but  one  degree  of  freedom;  that  is,  it  can  vary  along  the 
real  axis  only.  It  can  approach  the  limiting  position  from  two  pos- 
sible directions.  On  the  other  hand,  the  variable  z  =  x  -\-  iy  can 
be  said  to  have  two  degrees  of  freedom  since  x  and  y  are  independent 
variables.  The  variable  z  can  then  approach  its  limiting  position, 
not  only  from  two  possible  directions,  but  from  any  direction  in  the 
plane,  or  through  any  set  of  values  of  z  dense  at  a.  In  order  to 
affirm  that  f(z)  is  continuous,  we  must  be  able  to  say  that  the  same 
limiting  value,  namely  /(a),  is  obtained  if  z  is  allowed  to  approach 
a  through  every  possible  set  of  points  dense  at  a. 

If  a  is  a  boundary  point  of  a  region,  then  f{z)  is  said  to  be  con- 
tinuous at  a  if  f{z)  approaches  f{a)  through  every  set  of  inner  points 
of  the  region  dense  at  a.  We  say  that  a  function  is  continuous 
throughout  a  region,  whether  open  or  closed,  if  it  is  continuous  at 
each  point  of  the  region. 

We  can  establish  the  following  theorem  as  a  consequence  of  the 
definition  of  continuity. 

Theorem  I.  //  f(z)  is  continuous  at  an  inner  point  Zq  of  a  region 
S  and  if  f{zo)  7^  0,  then  there  exists  a  neighborhood  of  Zq  for  which 
f(z)  ^  0. 

Since  f{z)  is  continuous  at  z  —  zo,  we  have 

Lf(z)=f(zo); 

that  is,  for  an  arbitrarily  small  positive  number  e,  there  exists  a  posi- 
tive number  8  such  that 

l/(2)-/(2o)|<6       -     ^  (4) 

for  \z  —  Zo\  <8.  But  as /(20)  is  a  constant  different  from  zero,  we 
may  write 

\f(zo)-0\=A>0.  (5) 

By  taking  e  <  ^,  we  have  by  combining  (4)  and  (5) 

|/(0)-O|>|,        \z-Zo\<8.      ■ 
But  as  A  is  greater  than  zero,  this  relation  establishes  the  theorem. 


Art.  13]  CONTINUITY  .  35 

The  continuity  of  f{z)  =u-\-iv  with  respect  to  z  may  be  seen  to 
depend  upon  that  of  u{x,  y),  v(x,  y)  with  respect  to  x,  y,  as  stated 
in  the  following  theorem. 

Theorem  II.  The  necessary  and  sufficient  condition  that  f(z)  is 
continuous  at  z  =  a  is  that  u{x,  y),  v{x,  y)  are  both  continuous  in  x,  y 
at  the  corresponding  'point  {a,  b),  where  a  =  a-\-ib. 

This  result  follows  as  a  direct  consequence  of  Theorem  II  of  Art. 
12  and  the  definition  of  continuity. 

It  has  already  been  pointed  out  that  when  /(z)  is  continuous  at 
z  =  a,  we  may  write 

1/(2)  -/(«)!<€,  \z-a\<8,  (6) 

where  e  is  an  arbitrarily  small  positive  number  and  5  is  another  posi- 
tive number  depending  for  its  value  upon  e  and  a.  In  other  words, 
€  is  first  selected  as  small  as  we  choose  and  then  5  is  so  determined 
that  the  condition  given  in  (6)  is  satisfied.'  For  any  fixed  value  of 
€,  the  value  of  5  may  change  when  some  point  other  than  z  =  a  is 
taken  as  the  limiting  point.  Moreover,  for  any  particular  value  of 
z,  various  values  may  be  assigned  to  5.  If  Za  is  any  point  in  a  given 
region,  denote  by  5(0o)  the  value  of  5  corresponding  to  the  previously 
assigned  value  of  c.  Then,  for  this  given  e,  5(z)  is  a  function  of  z, 
made  up  of  the  totality  of  all  the  values  of  h{z)  as  z  takes  the  various 
values  in  the  given  region. 

Consider  now  the  function  5(z)  for  all  values  of  z  in  the  given 
region.  If  for  an  arbitrarily  chosen  e  >  0,  5(z)  has  a  lower  limit  5' 
different  from  zero,  we  say  that  the  function  /(z)  is  uniformly  con- 
tinuous in  the  given  region.  From  what  has  been  said,  it  will  now 
be  seen  that  an  essential  characteristic  of  uniform  continuity  is  that 
the  relation  (6)  is  satisfied  by  any  definite  value  5,  where  0  <  5  <  5', 
regardless  of  the  value  of  a. 

It  is  shown  in  the  theorj^  of  functions  of  a  real  variable  that  any 
function  that  is  continuous  throughout  a  closed  interval  is  uniformly 
continuous  in  that  interval.  The  corresponding  theorem  for  com- 
plex variables  may  be  stated  as  follows: 

Theorem  III.  If  a  function  f{z)  of  the  complex  variable  z  is  con- 
tinuous in  a  finite  closed  region  S,  then  it  is  uniformly  continuous  in 
that  region. 

To  prove  this  proposition,  we  shall  assume  the  contrary  to  be 
true  and  show  that  this  assumption  leads  to  a  contradiction.     The 


36 


DEFINITIONS  CONCERNING  FUNCTIONS 


[Chap.  II. 


assumption  is  then  that  the  function  f{z)  does  not  satisfy  the  defini- 
tion for  uniform  continuity  in  the  closed  region  S.  Since  f{z)  is 
continuous  at  every  point  within  S  or  upon  its  boundary,  we  know 
from  the  foregoing  discussion  that  for  a  fixed  but  previously  assigned 
value  of  e  there  is  associated  with  each  point  z  of  the  region  a  defi- 
nite number  8{z),  which,  however,  may  vary  with  the  point.  The 
function  8{z)  is  fully  defined  at  all  points  within  *S  or  upon  its  bound- 
ary. By  the  assumption  that  /(«)  is  not  uniformly  continuous,  we 
have  the  condition  that  the  lower  limit  of  8{z)  is  zero.  Inclose  the 
region  S  in  a  rectangle  by  drawing  lines  parallel  to  the  two  axes. 

Divide  this  rectangle  Ri  into  four 
equal  parts  by  again  drawing  lines 
parallel  to  the  axes.  In  the  part 
of  S  lying  in  at  least  one  of  these 
subdivisions,  say  Rz,  8{z)  must  have 
the  lower  limit  zero.  Divide  in  the 
same  way  R2  into  four  equal  parts; 
in  one  of  these  divisions,  say  R3, 
8(z)  has  the  lower  limit  zero.  Con- 
tinue this  process  indefinitely.  In 
X  the  limit  the  sequence  of  rectangles 
Ri,  R2,  Ri,  .  .  .  defines  a  definit'e 
point  z'  (Art.  12),  which  may  be  a 
point  within  or  upon  the  boundary  of  the  region  S.  In  any  case, 
since  >S  is  a  closed  region,  2'  is  a  point  of  S.  We  may  then  say  that 
there  is,  under  the  assumption  as  to  uniform  continuity,  at  least  one 
point  z'  of  the  given  region  such  that  in  every  neighborhood  of  2'  the 
lower  limit  of  the  5's  is  zero. 

The  given  function /(2)  is,  however,  continuous  for  2  =  2',  and  hence 
for  the  point  2'  there  exists  a  80  different  from  zero,  where  5o  =  8{z'), 
such  that  for  any  two  values  21,  22  of  the  variable,  for  which 

1  2i  -  2'  I  <  5o,  I  22  -  2'  I  <  5o, 


Y 

?.^ 

»V           ) 

/                 '                 •■.■-,v>7v.;.s( 

0 

X 

Fig.  12. 


we  have 


f(zO-f(z') 


<!' 


l/(^2)-/(2')|< 


Combining  these  inequalities,  we  have 

lM)-/(22)l<e; 


Art.  13.] 


Qs^ 


CONTINUITY 


37 


that  is,  the  oscillation  of  the  function  within  a  circle  of  radius  5o 
about  z'  can  not  exceed  the  arbitrarily  small  number  e.     If  we  now 

draw  another  circle  C  of  radius  ^  about  2'  as  a  center,  then  at  every 

point  within  C  the  given  function  j{z)  is  not  only  continuous  but  if 

a  circle  of  radius  -^  be  drawn  about  any  such  point  the  upper  limit 

of  the  oscillation  of  j{z)  within  this  circle  likewise  can  not  exceed  the 
arbitrarily  small  number  e.     Hence,  the  value  of  5  associated  with 

any  point  in  C  can  nob  be  less  than  -j^,  which  in  turn  is  greater  than 

zero.  If  the  point  z'  lies  upon  the  Y^ 
boundary  of  *S,  we  consider  only  that 
portion  of  the  region  bounded  by  the 
two  circles  which  hes  in  »S.  Conse- 
quently, we  can  now  say  that  the 
lower  limit  of  the  5's  for  all  points 
lying  within  C  can  not  be  zero  as 
assumed.  From  this  contradiction 
the  theorem  follows. 

From  the  definition  of  continuity 
of  /(z)  at  a  boundary  point  and  from 
the  foregoing  theorem,  we  conclude 
that  if  j{z)  is  continuous  at  each 
point  of  an  arc  C,  end  points  included,  of  the  boundary  of  a  closed 
region  <S,  then  we  have  for  any  point  Zo  of  C 

1/(2)  -/(20)  I  <e,  |z-ZoI  <5, 

where  the  values  of  z  correspond  to  inner  points  of  S  and  5  is  in- 
dependent of  Zo,"  that  is,  for  a  given  e  there  exists  a  5  that  satisfies 
the  required  conditions  equally  well  for  all  points  Zq  of  C,  end  points 
included.  We  say  then  that  /(z)  converges  uniformly  along  C. 
Expressed  in  terms  of  limits,  it  follows  that  /(z)  converges  uniformly 
along  the  arc  C  if  the  limit 

I'   f(z)=f(zo) 


O 


Fig.  13. 


exists  when  taken  over  inner  points  of  *S  for  each  point  Zq  of  C,  end 
points  included. 

We  are  able  now  to  state  the  following  theorem  concerning  uni- 
form convergence  along  an  arc. 


Iua^ 


38  DEFINITIONS  CONCERNING  FUNCTIONS  (Chap.  II. 

Theorem  IV.  //  f{z)  is  defined  for  a  closed  region  S  and  con- 
verges uniformly  along  an  arc  C  of  the  boundary  of  S,  then  f(t)  is  con- 
tinuous, where  t  denotes  the  values  of  z  on  C. 

Let  to  be  any  point  of  the  arc  C.  Then  from  the  definition  of 
uniform  convergence,  we  have 

\f(z)-f(U>)\<e,         \z-to\<8,  (1) 

where  z  is  confined  to  inner  points  of  S,  that  is  to  inner  points  within 
a  circle  70  of  radius  8  about  /o-  Let  ti  be  any  other  point  of  C  situ- 
ated within  7o.     Then  by  hypothesis,  we  have  also 

\f(z)-fiti)\<^,         \z-h\<8  (2) 

for  all  inner  points  of  S  within  a  circle  71  of  radius  5  about  ti.  The 
two  circles  overlap  and  inclose  common  inner  points  of  S.  For 
such  a  common  point  z'  of  2  we  have  from  (1) 

\fiz')-f(t)\<e, 
and  likewise  from  (2),  we  get 

l/(2')-/(t)l<t. 
Combining  these  two  inequalities,  we  obtain 

l/(t;)-/(^)l<2  6. 

But  ti  is  any  point  of  C  such  that  \'$i  —  f^\  <  8.  Hence,  /(fc)  is 
continuous  at  0o  and  therefore  for  all  values  of  2  along  C,  as  stated 
in  the  theorem. 

Theorem  V.  //  f(z)  is  continuous  in  a  finite  closed  region  S, 
then  there  exists  some  finite  number  M  such  that  \  f(z)  |  <  M  for  all 
values  of  z  in  S. 

To  estabhsh  this  theorem,  we  assume  the  contrary  to  be  true, 
namely,  that  there  exists  no  finite  number  M  answering  the  condi- 
tions of  the  theorem,  and  shall  prove  that  this  assumption  leads  to  a 
contradiction  of  the  given  hypothesis.  Inclose  the  given  region  in 
a  rectangle  by  drawing  lines  parallel  to  the  two  axes  and  divide  this 
rectangle  into  four  equal  parts.  In  at  least  one  of  these  subdivisions 
I  f{z)  I  exceeds  every  finite  value.  Let  Ri  be  such  a  subdivision. 
Divide  Ri  into  four  equal  parts  likewise  by  drawing  lines  parallel  to 
the  two  axes.  In  at  least  one  of  these  new  subdivisions,  say  R2, 
\f(z)  I  must  also  exceed  every  finite  bound.     Divide  R2  into  four 


Art.  13.]  CONTINUITY  39 

equal  parts  in  the  same  maimer  and  regard  this  process  as  continued 
indefinitely.     In  this  way  a  sequence  of  rectangles 

Rl,  Ri,  .  . ..  ,  Rn,  .  .  . 

is  obtained  satisfying  the  conditions  of  Theorem  IV,  Art.  12. 

In  the  limit  these  rectangles  define  a  point,  say  Zo,  and  as  the  given 
region  <S  is  a  closed  region  the  point  Zo  is  a  point  of  *S.  It  can  now 
be  said  that  in  every  deleted  neighborhood  of  Zo,  fiz)  exceeds  in  abso- 
lute value  every  finite  bound.     Consequently,   the  limit     L   f{z) 

can  not  be  said  to  exist;  because,  a  set  of  points  Zi,  Z2,  zs,  .  .  .  , 
z„,  .  .  .  ,  having  zo  as  a  limiting  point,  can  be  selected  so  that  as  z 
approaches  Zo  through  these  points,  the  function  |/(z)  |  exceeds  all 
finite  limits,  that  is,  becomes  infinite.  From  the  definition  of  con- 
tinuity, it  follows  then  that  /(z)  is  discontinuous  at  Zo.  This  con- 
clusion is  a  contradiction  of  the  hypothesis  set  forth  in  the  theorem 
and  from  this  contradiction  the  theorem  follows. 

Theorem  VI.  If  f(z)  is  continuous  in  a  finite  closed  region  S, 
then  I  /(z)  I  has  a  finite  upper  limit  in  S. 

From  Theorem  V  we  know  that  |/(z)  |  has  a  finite  upper  bound; 
that  is,  there  exists  a  finite  number  M  such  that  |  /(z)  |  <  M.  The 
function /(z)  is  then  represented  by  points  lying  within  a  circle  Ci  about 
the  origin  having  the  radius  M.  The  present  theorem  asserts  that 
there  exists  a  circle  of  radius  equal  to  or  less  than  M  such  that  there 
are  values  of  /(z)  represented  by  points  indefinitely  close  to  or  upon 
this  circle  but  not  without  it.  Divide  the  radius  M  of  the  circle  Ci 
into  10  equal  parts  by  drawing  about  the  origin  concentric  circles 
having  the  radii 

M     2_M  9M 

lO'      10  '  '  •  •  '     10  ' 

Among  these  circles,  including  Ci,  there  is  a  smallest  one  such  that 
no  values  of  /(z)  are  represented  by  points  lying  outside  of  it.  Sup- 
pose the  next  smaller  circle  C2  has  the  radius  -h^,  where  h  may 

have  any  one  of  the  values  0,  1,  2,  .  .  .  ,  9.  li  ki  =  0,  the  circle  d 
becomes  a  point,  namely  the  origin.  Between  C2  and  the  next  larger 
circle  insert  10  other  concentric  circles  having  respectively  the  radii 

Mf      M      Mf  ,2M  kiM      9M 

10   "^102'      10   "^  102'  •  •  •  '    10   "^  102* 


40 


DEFINITIONS  CONCERNING  FUNCTIONS 


(Chap.  II. 


Among  these  circles,  including  the  circle  of   radius       ^         — , 

there  is  likewise  a  smallest  such  that  no  values  of  f{z)  are  represented 
by  points  exterior  to  it.     Let  the  circle  next  smaller  than  that  circle 

be  Cs,  having  the  radius  -7^  +  -j^^ ,  where  again  k^  may  take  any 

one  of  the  values  0,  1,  2, 
get  a  sequence  of  circles 


10 


102 


9.     Continuing  in  this  manner,  we 


C2,  C3,  C4, 


having  respectively  the  radii 
kjM     kiM  ,  k^M 
10  ' 


10 


102 


kiM      kjM 

~10"  "•"  lo^" 


hM 


W 


This  sequence  of  numbers  satisfies  the  conditions  of  Theorem  VI, 
Art.  12,  and  hence  has  a  limit,  say  G.  This  limit  is,  however,  the 
upper  Umit  required,  for  the  sequence  determines  the  radius  of 
the  least  circle  such  that  no  values  of  /(«)  are  represented  by  points 
without  it. 
We  may  now  state  the  following  theorem: 

Theorem  VII.     If  fiz)  is  continuous  in  a  finite  closed  region  S,  then 
I  f{z)  I  attains  its  upper  limit  in  S. 

This  theorem  is  equivalent  to  saying  that  every  function  f{z) 

which  is  continuous  throughout  a 
finite  closed  region  is  such  that 
I  f(z)  I  possesses  a  maximum  value. 
Construct  the  rectangle  (a:,  61, 
Ci,  di),  Fig.  14,  inclosing  the  given 
region  S  for  which  the  function  is 
defined.  Divide  this  rectangle  into 
four  equal  parts  by  drawing  lines 
parallel  to  the  two  axes.  Consider 
only  those  rectangles  that  contain 
at  least  one  point  of  the  given 
region  *S.  By  Theorem  VI,  |  f(z)  \ 
has  in  >S  a  finite  upper  hmit.  De- 
note this  upper  Umit  by  G.  Then  in  some  one  of  the  subdivisions  of 
the  original  rectangle,  say  (02,  61,  C2,  c^),  |  f{z)  \  must  have  the  upper 
limit  G.  Divide  the  rectangle  (02,  &i,  Cs,  d^)  likewise  into  four  equal 
parts  as  shown  in  the  figure;  some  one  of  these  new  divisions,  say 
(as,  63,  Cs,  di)  must  contain  such  values  of  f(z)  that  |  f{z)  \  has  the 


Art.  13.]  CONTINUITY  41 

upper  limit  G.  Consider  this  process  of  subdivision  as  continued 
indefinitely.  By  each  division,  some  one  of  the  subdivisions  must 
be  such  that  |  j{z)  \  has  the  upper  limit  G  for  the  values  of  z  in  it. 
By  this  method  of  subdivision  each  rectangle  that  is  chosen  is 
situated  within  all  of  those  that  have  entered  previously  into  con- 
sideration. The  dimensions  of  the  rectangles  have  the  limit  zero 
and  the  sequence  of  rectangles  defines  in  the  limit  a  definite  point, 
which  we  may  denote  by  Zq.  Since  f{z)  is  continuous  for  z  =  zq,  we 
have 

Lf(z)=f(zo).  (1) 

But  as  the  limit  of  the  absolute  values  of  a  function  is  the  absolute 
value  of  the  limit,  we  have  also 

L    1/(^)|  =  IM)|.  (2) 

In  every  neighborhood  of  Zo  the  upper  limit  of  |  f{z)  \  is  G.  The  limit 
of  I /(z)  I,  as  z  approaches  Zo,  can  not  then  be  greater  than  G.  We  shall 
now  show  that  this  limit  can  not  be  less  than  G;  for,  suppose  it  is 
G  —  A,  where  A  9^  0.    We  then  have  for  an  arbitrarily  chosen  posi- 

live  number  e,  say  e  —  -^ ,  another  positive  number  5  such  that 

\\f(z)\-{G-A)\<e,         |z-Zo|<5; 

that  is,  for  all  values  of  z  within  the  circle  having  zo  as  center  and  8 
as  radius  |  /(z)  |  differs  from  G  —  A  hy  less  than  e.  This  is  a  con- 
tradiction to  the  foregoing  conclusion  that  the  upper  limit  of  /(z)  in 
every  neighborhood  of  Zo  is  G.     Hence,  the  Umit  L  |  /(z)  |  can  not 

be  less  than  G,  and  as  it  must  exist  and  can  not  exceed  G,  it  must 
be  equal  to  G.    We  have  then 

L    \f(z)\  =  G.  (3) 

By  comparing  (2)  and  (3),  we  obtain 

and  the  theorem  is  established. 


42  DEFINITIONS  CONCERNING  FUNCTIONS  [Chap.  H. 

EXERCISES 

■y    1.  Classify  the  following  functions:  _ 

(a)  3x»   +7x2 +19,        .A-^i^  »-^^ 

(6)  5x»   +2x*+6x     +3,    tA-^^^     .  -,_//•/ 

(c)  2x-»  +  7xJ  +3X-1  +  4,   ^.*:t;^  ^uw/**'-^  r**^~ 

(d)  tan  X,  cos  X,  log  x.       ^.<t..*;«>4*wip4s 

2.  Show  that  1  is  the  limit  of  the  sequence 

f>    5»     7>     5>    •    •    •    • 

3.  Given  /(«)  =  (x<  +  y<  -  6  xV)  +  i(4  x»i/  -  4  xy>').     Find  the  Umit  of  /(«) 
as  z  approaches  2  +  3  t. 

4.  Show  that  the  sequence  of  circles  given  by  the  equation 


(^-^r 


+  y'  =  Z5.     «  =  1, 2, 3, 


defines  in  the  limit  one  and  only  one  point,  namely  the  origin. 

2  XXI 
6.  Show  that  /(z)  =  log  (x*  +  w*)  +  i  arc  tan  —. — —.  is  continuous  for  finite 

x^  —  y^ 

values  of  z  =  X  +  iy  ?i  0. 

6.  Show  that  an  integral  rational  function  of  z  is  uniformly  continuous  in  any 
given  finite  region. 

z*  +  1 

7.  Given /(z)  =  Y——r.     Show  that  | /(z)  |  has  a  finite  upper  limit  in  the  region 

bounded  by  the  circle  x*  +  y*  =  \.     Find  thb  upper  limit. 

8.  Find  the  limit  of  the  sequence 

a\  02,  .  .  .  an,  .  .  . 
where  ""  =  2^ +  *  [^  "(i)  ]' 

9.  Given  a  closed  region  S  in  which  /(z)  is  continuous.  Show  that  for  an 
arbitrarily  chosen  positive  number  e  there  exists  another  positive  number  5  such 
that 

\f(.Zi)  -fiz2)\<e, 

for  I  Zi  —  Z2 1  <S,  Zi,  Zi  being  points  of  S. 


CHAPTER  III 

DIFFERENTIATION  AND  INTEGRATION 

14.  Differentiation;  analytic  function.  The  definition  of  the 
derivative  of  a  function  /(z)  of  a  complex  variable  is  identical  in  form 
with  the  definition  of  the  derivative  of  a  function  of  a  real  variable. 
Let  z  be  a  variable  point  in  the  neighborhood  of  Zo-     Put 

A2  =  z  —  Zo, 
and 

^w  ^  f(z)  -  /(zo)  =  /(zo  +  Az)  -  /(zo). 
If  the  limit 

L  ^=  L  /^-^  +  Az)  - /(zq) 
Az=o  Az     Az=o  Az 

exists,  we  say  that  this  limit  is  the  derivative  of  /(z)  at  the  point  Zq. 
The  derivative  is  then  the  limit  of  the  ratio  of  the  two  complex  vari- 
ables Aw  and  Az.  From  the  properties  of  limits  already  discussed, 
it  will  be  seen  that  to  have  this  limit  exist,  the  same  limiting  value 
must  be  obtained  independently  of  the  path  along  which  z  approaches 
Zo.  Moreover,  the  same  limiting  value  must  be  obtained  if  we  select 
any  possible  set  of  values  for  z  dense  at  Zo  and  let  z  approach  Zo 
through  these  values.  The  value  of  the  limit  must  therefore  be 
independent  of  the  amplitude  of  Az  as  z  approaches  Zo.  We  shall 
make  use  of  the  same  symbols  as  in  the  calculus  of  a  real  variable  to 
denote  a  derivative;  for  example,  the  derivative  oi  w  =  f(z)  with 
respect  to  z  is  denoted  by  any  one  of  the  symbols: 

The  general  laws  of  differentiation  for  real  variables  can  be  ex- 
tended without  modification  to  functions  of  a  complex  variable, 
since  they  depend  upon  the  general  laws  of  limits,  which  hold  equally 
well  in  both  fields.     For  example,  we  have 

(a)  D^{cw)  =  cDzW,  • 

(b)  Dz(wi  H-  W2)  =  DzWi  +  DzWi, 

(c)  Dz{wi '  w-i)  =  WiDzVh  +  WiDzWi, 

43 


44  DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 

._  -^  /wi\  ^  VhP.Wi  -  WiDtWj^ 

'\Wi)  Wi^ 

(c)  D,f(w)  =  Du,f(w) .  D^{w),  etc. 

It  is  suggested  that  the  student  deduce  the  foregoing  laws  for  differ- 
entiation directly  from  the  definition  of  a  derivative.  As  with 
functions  of  a  real  variable,  the  continuity  of  the  function  is  a  neces- 
sary condition  for  the  existence  of  the  derivative. 

The  differentials  dw,  dz  may  be  defined  in  precisely  the  same 
manner  that  differentials  are  defined  in  the  calculus  of  real  variables. 
For  this  purpose,  suppose  w  and  z  are  expressed  in  terms  of  a  third 
common  variable  t.    We  have  then 

w  =  w{t),        z  =  z{t),        where    w=f{z). 

If  we  differentiate  with  respect  to  the  common  variable  t,  we  have 

Diw  =  DJ{z)  •  D,z. 

As  the  derivatives  DtW,  DiZ  with  respect  to  the  common  variable 
enter  homogeneously  into  the  above  identity,  we  define  the  differ- 
entials dw,  dz  as  numbers  equal  to  or  in  proportion  to  these  deriva- 
tives and  write 

dw  =  Dzf{z)  dz. 

The  parametric  representation  introduced  above  gives  us  some 
advantage  in  certain  discussions.  For  example,  we  shall  frequently 
have  occasion  to  introduce  the  condition  that 

X  =  <i>{t),        y  =  ^(0, 

where  <i>,  ^  are  real  functions  of  a  real  variable  t  and  possess  contin- 
uous first  derivatives  with  respect  to  t.  We  then  have  from  the 
definition  of  a  differential 

dx  =  <i>'{t)  dt,        dy  =  ^'(0  dt.  (1) 

The  connection  between  the  real  functions  <t>{t),  \p{t)  and  the  com- 
plex variable  z  is  given  by  the  equation  z  =  x  -\-  iy.  We  have 
therefore 

dz  =  Dtz  dt 

=   W{t)-\-irl^t)\dt. 

Replacing  <i>'{t)  dt,  ^'(t)  dt -by  their  values  as  given  in  (1)  we  have 

dz  =  dx  -\-  i  dy. 


Akt.  14.]  ANALYTIC  FUNCTION  45 

The  higher  derivatives  and  higher  differentials  follow  the  same 
laws  as  in  the  calculus  of  real  variables  and  the  same  symbols  are 
used  to  represent  them. 

As  we  have  already  pointed  out  (Art.  11),  the  general  definition 
of  a  function  does  not  impose  upon  the  functional  correspondence 
such  special  properties  as  continuity,  differentiability,  etc.  To  say 
that  /(z)  is  a  function  of  the  complex  variable  z  asserts  nothing  further 
than  that  f{z)  depends  upon  z  in  such  a  manner  that  for  each  value 
given  to  z  there  is  thereby  determined  a  definite  value  or  set  of  values 
of  the  function  /(z) .  We  make  a  substantial  advance  when  we  can 
ascribe  to  a  function  the  properties  of  continuity  and  differentia- 
bility. The  functions  to  be  considered  in  this  volume  possess  for  the 
most  part  both  of  these  properties. 

If  a  given  single-valued  function  f{z)  has  a  uniquely  determined 
derivative  at  the  point  a  and  at  every  point  in  the  neighborhood  of 
a,  then  z  =  a  is  called  a  regular  point  of  f{z).  By  some  authors, 
the  function  is  said  to  be  analytic  at  z  =  a  and  by  others  it  is  called 
holomorphic  at  this  point.  We  shall,  however,  reserve  these  terms 
for  other  uses. 

A  point  in  every  deleted  neighborhood  of  which  there  are  regular 
points  but  which  is  itself  not  a  regular  point  is  called  a  singular  point 
of  the  given  function.  ^  .  '      "^  "  J^ "    ^ 

If  every  point  of  a  given  region  *S  is  a  regular  point  of  a  single- 
valued  function  j{z),  then  /(z)  is  said  to  be  holomorphic  in  *S.  It 
should  be  borne  in  mind  throughout  this  and  the  succeeding  chapters 
that  we  have  defined  a  region  to  be  a  continuum  of  inner  points; 
hence,  it  is  understood  that  a  region  does  not  include  its  boundary 
points  unless  so  specified.  We  shall  speak  of  a  function  f{z)  as  being 
an  analytic  function  of  2  if  it  is  holomorphic  in  at  least  some  region 
S  with  the  possible  exception  of  certain  singular  points  which  do  not 
interrupt  the  continuity  of  S.  It  is  always  possible  then  to  join  any 
two  regular  points  of  *S  by  a  continuous  curve  which  lies  wholly  within 
S  and  which  does  not  pass  through  a  singular  point.  A  more  pre- 
cise definition  of  analytic  functions  will  be  given  in  Chapter  VII. 

From  the  definition  of  a  function  which  is  holomorphic  in  a  region, 
we  have  at  once  the  following  general  properties.  Given  two  func- 
tions J{z),  <t){z),  each  holomorphic  in  a  region  S;  then  it  follows  that 
inS: 

!•  /(2)  +  <^(2)  is  holomorphic, 

2.  f{z)  •  (f>{z)  is  holomorphic, 


46  DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 

3.  A~{  is  holomorphic,  except  for  those  values  of  z  for  which  <t>{z)  =  0. 
4>{z) 

4.  //  Wo  is  a  regular  point  of  f{w),  and  Zo  is  a  regular  point  of 
w  =  <t>(.z),  where  <l>{zo)  =  Wa,  then  Zq  is  a  regular  point  of  the  function 
f  \<t>iz)l  considered  as  a  function  of  z. 

From  these  properties,  it  follows  that  every  rational  integral 
function  of  z  is  an  analytic  function,  holomorphic  in  the  finite  region 
of  the  complex  plane.  Since  every  rational  function  is  holomorphic, 
except  at  most  at  a  finite  number  of  points  where  the  denominator 
is  zero,  it  also  is  an  analytic  function. 

15.  Line-integrals.  It  was  pointed  out  in  the  last  article  that 
the  definition  of  the  derivative  of  a  function  of  a  complex  variable 
involves  a  more  complicated  limit  than  the  corresponding  definition 
in  the  case  of  functions  of  a  real  variable.  A  similar  generalization 
is  necessary  in  the  discussion  of  integration,  in  that  we  must  in  gen- 
eral take  into  account  the  path  along  which  the  integral  is  to  be 
taken.  In  the  case  of  functions  of  a  real  variable,  the  independent 
variable  x  can  pass  continuously  from  any  value  Xi  to  some  other 
value  X2  along  only  one  path,  namely,  by  passing  through  the  inter- 
mediate values. along  the  X-axis.  In  the  case  of  functions  of  a  com- 
plex variable,  the  independent  variable  z  can  pass  from  a  value-  zi 
to  another  value  Z2  by  any  number  of  different  paths.  Consequently, 
*  the  definition  of  a  definite  integral  between  two  values  of  z  can  have 
a  significance  only  when  we  consider  the  path  by  which  z  passes  from 
the  one  value  to  the  other. 

As  the  subject  is  not  always  considered  in  elementary  text-books 
on  calculus,  we  shall  now  define  a  line-integral  and  discuss  some  of 
the  more  general  properties  of  such  integrals.  Among  other  things, 
we  shall  show  that  the  integral  of  a  function  of  a  complex  variable 
taken  over  a  given  path  may  be  expressed  in  terms  of  line-integrals 
of  functions  of  the  real  variables  x,  y  taken  over  the  same  path. 

In  the  calculus  of  real  variables  a  definite  integral  is  defined  as 
the  limit  of  a  sum;  that  is 

rV(x)dr=  L  T/(?.)A^.  (1) 

Stated  in  words:  the  portion  of  the  X-axis  between  a  and  6  is  divided 
into  n  parts,  the  length  AkX  of  each  division  is  multiplied  by  the  value 
of  the  function  at  some  arbitrary  point  ^k  in  that  division  and  the 
limit  of  the  sum  of  these  products  is  taken,  as  the  number  of  such  divi- 


Aht.  15]  LINE-INTEGRALS  47 

sions  increases  without  limit  while  the  length  of  the  divisions  simulta- 
neously approaches  zero.  It  is  of  importance  to  observe  that  the  n 
divisions  between  the  limits  of  integration  are  taken  along  the  X-axis 
and  each  of  these  divisions  is  multiplied  by  a  value  of  the  func- 
tion at  a  point  on  the  same  axis.  Suppose  instead,  these  divisions 
and  the  points  at  which  the  functional  values  are  to  be  used  as  mul- 
tipliers are  taken  along  some  curve  C,  called  the  path  of  integration, 
the  function  with  whose  values  we  are  concerned  now  being  a  func- 
tion of  the  two  variables  x,  y.  Let  the  functional  value  at  a  point 
on  the  curve  in  each  division  be  multiplied  by  Ax,  which  is  the  orthog- 
onal projection  upon  the  X-axis  of  the  division  of  the  curve.  The 
limit  of  the  sum  of  these  products  as  Ax  approaches  zero,  that  is 
as  the  number  of  divisions  increases  indefinitely,  is  a  line-integral  of 
the  given  function  along  the  path  C.  This  curve  may  lie  in  the 
XF-plane,  or  in  case  we  have  a  function  of  three  real  variables,  the 
path  of  integration  may  be  a  curve  in  space.  In  the  particular 
applications  to  be  made  of  integrals  in  the  present  volume  the  path 
of  integration  will  always  be  a  plane  curve.  Any  rectifiable  curve, 
that  is  any  curve  having  a  definite  length,  may  be  taken  as  the  path 
of  integration.  However,  as  there  is  a  certain  element  of  arbitra- 
riness in  the  choice  of  the  path  of  integration,  we  shall  avoid  certain 
complications  in  the  discussions  to  follow  by  taking  as  that  path  a 
curve  that  may  be  broken  up  into  a  finite  number  of  divisions,  each 
of  which  is  either  a  rectilinear  segment  parallel  to  one  of  the  coordi- 
nate axes  or  else  has  the  property  that  it  is  determined  by  a  function 
y  =  <l>(x),  where  <^(x)  and  its  inverse  function  x  =  \l/{y)  are  single- 
valued  and  have  first  derivatives  that  are  continuous  except  at  most 
at  the  end  points,  at  which  points  they  may  become  infinite.  Such" 
a  curve  is  monotone  by  segments  and  for  convenience  will  be  desig- 
nated as  an  ordinary  curve,*  whenever  a  special  name  is  necessary 
for  the  sake  of  clearness.  However,  in  the  present  volume  only 
ordinary  curves  will  be  employed. 

Let  AB  be  one  of  the  finite  number  of  divisions  of  which  an  ordi- 
nary curve  C  (Fig.  15)  is  composed.  Let  the  coordinates  of  the 
points  A,BhQ  (xo,  i/o)  and  (x„,  y„),  respectively.  Divide  the  axe  AB 
into  n  parts  by  the  insertion  of  w  —  1  points  pi,  p2  •  •  •  ,  pt  .  .  .  , 
p„_i,  whose  coordinates  are  (xi,  yi),  {x^,  y^),  .  .  .  (xk,  yu),  -  .  . 
(x„-i,  yn-\),  respectively. 

*  Compare:  Pringsheim,  Encyklopdiie  der  Math.  Wiss.,  II,  A  1,  p.  22;  also 
Dodd,  BuU.  of  Univ.  of  Tex.,  No.  222,  March,  1912. 


48 


DIFFERENTIATION  AND  INTEGRATION 


[Chap.  III. 


Select  at  pleasure  a  point  {^k,  tj*)  upon  each  arc  {pk-i,  Pk),  k  =  1, 
2,  .  .  .  ,  n.  Having  given  a  function  F{x,  y)  which  is  continuous  in 
X  and  y  together  along  the  curve  C,  form  the  sum  of  the  products 

of  the  subintervals 

Ajtx  =  Xk  —Xk-\ 

and  the  values  of  the  given  function 
F  (x,  y)  at  the  points  (^t,  -rji).  Fi- 
nally, consider  the  limit  of  this  sum 
as  the  number  of  subintervals  AkX 
between  A  and  B  is  increased  in- 
definitely, while  at  the  same  time 
the  length  of  each  interval  ap- 
proaches zero,  namely,  the  limit 


Fig.  15. 


(2) 


If  this  limit  exists,  it  is  called  a  line-integral  *  of  F{x,  y)  along  the 
given  curve  C  between  tlie  limits  A  and  B.  Such  an  integral  is 
represented  by  the  symbols 

/  F{x,y)dx,  \'^F{x,y)dx,  /     F(x,y)dx. 

In  defining  a  line-integral,  we  took  the  limit  of  a  sum  of  products 
formed  by  multiplying  FiXk,  TQt)  by  the  projection  of  the  arc  (pt- 1,  p*) 
upon  the  X-axis.  By  taking  the  projection  of  this  arc  upon  the 
y-axis,  we  may  define  in  an  analogous  manner  the  line-integral 


L 


F(x,  y)  dy. 


It  will  be  observed  that  the  ordinary  definite  integral  is  merely  a 
special  case  of  a  line-integral,  namely,  where  one  of  the  axes  of  co- 
ordinates is  taken  as  the  path  of  integration. 

The  existence  of  the  limit  defining  a  line-integral  may  be  made  to 
depend  upon  that  defining  an  ordinary  definite  integral.  Let  us 
assume  that  F{x,  y)  is  a  continuous  function  of  the  two  variables 
z,  y  together  along  the  path  of  integration.  Let  an  arc  AB  of  the 
curve  y  =  <i>{x)  be  selected  as  the  path  of  integration  (Fig.  16).  For 
the  present,  we  shall  also  restrict  the  discussion  to  the  case  where 


*  Sometimes  called  also  a  curvilinear  integral. 


Abt.  15. 


LINE-INTEGRALS 


49 


no  line  parallel  to  the  F-axis  cuts  this  arc  in  more  than  one  point. 
We  may  replace  y  by  (f>{x)  and  write 

F{x,y)=Fix,4>(x))=f{x),  (3) 

where  f{x)  is  a  continuous  function.     The  limit  considered  in  (2) 
then  becomes  • 


Since /(x)  is  a  continuous  function, 
this  limit  exists  and  defines  the 

definite  integral*    I    fix)  dx,  and 

we  have 

rf(x)dx=  f   F{x,<f>{x))dx,   (4) 
J  a  Jab 


h,Vk 


Fig.  16. 


(5) 


in 


where  a,  h  are  the  projections  of  A, 
B,  respectively,  upon  the  X-axis. 

Consequently,  the  line-integral    /  F(x,  y)  dx  not  only  exists  when 

Fix,  y)  is  continuous  in  x,  y,  but  we  may  write 

J  Fix,  y)dx  =    I   fix)  dx. 
c  Ja 

It  will  be  observed  that  the  integral     I  Fix,  y)  dx  depends 

general  upon  the  curve  C  as  well  as  upon  the  function  Fix,  y) ;  for, 

taking  x,y,zsis  the  space  coordinates  of  a  point,  the  integral  /   fix)  dx 

is  represented  by  the  shaded  area  (Fig.  16)  under  the  curve  z  =  fix). 
This  area  is  the  projection  upon  the  XZ-plane  of  the  area  upon  the 
cylinder  perpendicular  to  the  XF-plane  through  the  path  of  inte- 
gration y  =  0(x),  and  underneath  the  curve  of  intersection  of  this 
cylinder  and  the  surface  z  =  Fix,  y).  As  the  path  of  integration 
y  =  (j){x)  changes,  the  cylinder  changes  and  of  course  the  projected 
area  may  change. 

In  the  discussion  thus  far  we  have  considered  only  the  case  where 
the  curve  y  =  <t>ix)  is  cut  by  a  line  parallel  to  the  F-axis  in  but  a 

"    *  See  Townsend  and  Goodenough,  First  Course  in  Calculus,  p.  177,  Art.  80. 


50 


DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 


single  point.    It  may  happen  that  such  a  line  may  meet  the  given 
arc  AB  in  more  than  one  point,  say  at  the  points  yi,  ?/2,  .  .  .  ,  as 

shown  in  Fig.  17.  In  such  a  case  the 
given  arc  should  be  divided  into 
several  portions  such  that  each  por- 
tion satisfies  the  required  condition. 
As  the  path  of  integration  is  an  arc 
of  an  ordinary  curve,  the  number 
of  such  subdivisions  is  always  finite. 
In  the  case  shown  in  the  figure, 
the  arc  AB  may  be  decomposed 
into  the  arcs  AD,  DE,  EB  where 
each  satisfies  the  necessary  condi- 
tion.    We  may  therefore  write 


Fig.  17. 


f  F{x,y)clx=  f   F{x,y)dx+  f   F{x,y)dx+  f  F(x,y)dx. 
Jab  J  ad  Jde  Jeb 


(6) 


If  P,  Q  are  two  real  functions  of  x,  y,  we  shall  understand  by  the 
line-integral 


(7) 


r"''''Pdx-\-Qdy,     or       \Pdx-\-Qdy, 

the  sum  of  the  two  line-integrals 

Pdx,  I         Qdy. 

From  what  has  been  said,  it  follows  that  these  integrals  exist  if  along 
C  the  functions  P,  Q  are  continuous  m.  x,  y  taken  together. 

It  is  frequently  convenient  to  change  the  independent  variables 
x,  2/  so  as  to  express  the  equation  of  the  path  of  integration  in  a  para- 
metric form.    For  example,  suppose  we  have 


a;=^i(0,        y=^2(0, 


(8) 


where  ^i(0,  ^2(0  a-re  continuous  functions  of  the  real  variable  t 
having  continuous  single-valued  first  derivatives. 

As  the  point  (x,  y)  varies  from  A  to  B  along  the  given  path  of  inte- 
gration, suppose  t  varies  from  U  to  tn.  Corresponding  to  the  divisions 
iVk-hVk)  of  the  arc  AB,  we  have  the  increments  A^f  =  tk  —tk-\, 
A;  =  1,  2,  .  .  .  ,  n.    By  the  law  of  the  mean,  we  have  then  from  (8) 

Xk  -  Xk-i  =  ^i'(<*')  •  {tk  -  tk-l), 


A(^^ 


/ 


Art.  15.]  LINE-INTEGRALS  51 

where  tk  lies  between  tk-\  and  h.  Corresponding  to  U  there  is  a  point 
(^ifc>  7)*)  of  the  arc  (pa;-i,  pt).  Hence,  if  we  have  given  a  continuous 
function  Fix,  y),  we  may  write 

n  n 

i=l  *=1 

Passing  to  the  limit  as  n  becomes  infinite,  we  have  from  the  definition 
of  a  line-integral 

f  P{x,  y)  dx  =  r  Pf^i  (0,  ^2(0  \^i'(t)  dt.  (9) 

%/  AB  Ota 

In  a  similar  manner,  we  may  show  that  if  Q{x,  y)  is  continuous  in 
X,  y  together,  we  have 

f  Q{x,  y)  dy  =  r  Q{^i(0,  Mt)  l^2'{t)  dt.  (10) 

*J  AB  »/fo 

For  the  general  form  of  the  line-integral  as  given  in  (7),  we  have 
then 


f    rdx-{-Qdy=   rV.*/(«)  +0-^2'(«)|c?«. 


(11) 


The  integrals  in  the  second  members  of  (9),  (10),  (11)  are  ordinary 
definite  integrals.  From  the  relations  expressed  in  these  equations, 
the  laws  of  operation  with  line-integrals  may  be  deduced  from  those 
of  ordinary  definite  integrals.  The  following  consequences  of  these 
relations  are  to  be  especially  noted. 

1.  The  law  for  the  change  of  variable  in  ordinary  definite  integrals 
applies  likewise  to  the  more  general  case  of  line-integrals. 

2.  The  integrals 

f  Pdx-\-Qdy,  f  Pdx  +  Qdy 

Jab  Jba 

have  the  same  numerical  value,  hut  are  opposite  in  sign. 

3.  //  Zo  is  any  point  upon  the  path  of  integration  AB,  then 

f  Pdx  +  Qdy=  f  Pdx  +  Qdy-\-  f  Pdx  +  Qdy. 
Jab  Jazo  Jz^b 

The  function  to  be  integrated  may  involve  a  parameter  in  addition 
to  the  variables  x,  y.     It  is  sometimes  desirable  to  be  able  to  differ- 


62  DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 

entiate  such  an  integral  with  respect  to  the  parameter.    Suppose 
we  have  given  the  line-integral 


Jab 


P{x,  y,  a)  dx, 
i 

where  P{x,  y,  a)  is  continuous  in  x,  y,  a  taken  together  and  where  the 

path  of  integration  AB  is  independent  of  a.     Suppose  also  that  — 

exists  and  is  hkewise  continuous  in  x,  y,  a.     We  ma^  then  show  that 

lln.y.a)^=fj^^<i^.  (12) 

This  result  follows  as  a  consequence  of  (9).    We  have 

^.£/^'''^'''^'^^^X"^^^'^^^'  ^2(0,  al^x'(Od«.         (13) 

dP 

Since  -r-  exists  and  is  continuous  in  (t,  a),  we  have  * 
da 

ifjl^M.  Mt),  ai*AOrf*  =  j;'-5^1M^^M)^*,'(<)*.  (14) 

This  last  integral  is  by  (9)  equal  to  /     — ^^'  ^'  ^^  dx. 

Jab        oa 

Hence,  we  have  from  (13)  and  (14)  the  required  relation  as  stated  in 

(12). 

The  path  of  integration  may  be  a  closed  curve,  that  is,  it  may  be 
the  boundary  of  a  given  region,  in  which  case  the  limits  of  integration 
are  represented  by  the  same  point  of  the  plane.  There  is  still  a 
choice  of  direction  in  which  the  integral  is  to  be  taken.  We  say  that 
it  is  taken  in  a  positive  sense  with  respect  to  the  region  bounded,  if 
it  is  so  taken  that  this  region  hes  always  to  the  left  of  the  observer 
as  he  proceeds  along  the  curve  in  the  direction  in  which  the  integral 
is  taken. 

Often  the  boundary  of  a  region  consists  of  two  or  more  closed 
curves.  For  example,  the  region  S  in  Fig.  18  is  bounded  by  the 
curve  M  and  the  circles  1,  2,  3.  We  may  speak  of  M  as  the  outer 
portion  of  the  boundary  and  of  the  circles  1,  2,  3  as  inner  portions 
of  the  boundary.  A  region  is  said  to  be  simply  connected  if  every 
closed  curve  in  it  forms  by  itself  a  complete  boundary  of  a  portion  of 
the  given  region.     A  region  that  does  not  satisfy  the  definition  of 

*  See  Gouraat-Hedrick,  Mathematical  Analysis,  Art.  97. 


Akt.  15.] 


LINE-INTEGRALS 


53 


a  simply  connected  region  is  called  a  multiply  comiected  region. 
The  region  S  shown  in  Fig.  18  is  a  multiply  connected  region,  since 
it  is  possible  to  have  a  closed  curve  surrounding  one  of  the  circles  that 
does  not  of  itself  form  a  complete  boundary  of  a  portion  of  the  region 


O 


Fl 


-^X 


O 


^X 


Fig.  18. 


Fig.  19. 


inclosed.  In  a  simply  connected  region,  a  curve  between  two  points 
may  always  be  changed  by  continuous  deformation  into  any  other 
curve  between  those  points,  both  curves  lying  completely  in  the  re- 
gion considered;  this  is  not  true  for  a  multiply  connected  region. 

A  finite,  multiply  connected  region  may  be  changed  into  a  simply 
connected  region  by  drawing  from  each  inner  portion  of  the  bound- 
ary a  line  connecting  it  with  the  outer  portion.  A  line  joining  two 
points  of  the  boundary  is  called  a  cross-cut,  and  we  shall  so  choose 
the  line  that  it  will  neither  intersect  itself  nor  other  cross-cuts.  For 
example,  the  multiply  connected  region  S  shown  in  Fig.  18  may  be 
changed  into  a  simply  connected  region  by  connecting  the  circles  1. 
2,  3  with  the  boundary  curve  M  by  cross-cuts  as  indicated  in  Fig.  19. 
It  serves  the  purpose  equally  well  to  connect  each  of  the  circles  with 
some  one  other  circle  by  means  of  cross-cuts  and  one  of  them  with 
the  outer  boundary  M.  Thus  in  Fig.  19  we  might  have  joined  circle 
2  to  circle  3  and  to  circle  1  and  then  have  joined  any  one  of  the 
three  circles  to  the  contour  M.  The  boundary  points  connected  by  a 
cross-cut  may  be  distinct,  as  in  the  case  of  the  cross-cuts  joining  the 
separate  curves  constituting  a  portion  of  the  boundary  in  Fig.  19  or 
the  cross-cut  a,  6  in  Fig.  20;  or  we  may  have  the  two  boundary 
points  coincident  as  in  the  case  of  the  cross-cut  drawn  from  the  point 
c  and  returning  to  the  same  point,  in  which  case  the  cross-cut  is  a 
closed  curve. 


54  DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 

The  notion  of  a  path  of  integration  may  now  be  extended  so  as  to 

include  the  boundary  of  a  multiply  connected  region;    for,  having 

made  the  region  simply  connected  by  the  introduction  of  cross-cuts, 

yi  the  integral  may  be  taken  along  the 

contour  of  this  simply  connected 
region  including  the  cross-cuts. 
For  example,  in  Fig.  19  the  arrows 
indicate  the  positive  direction  of 
the  integral  with  respect  to  the 
region  S'.  It  will  be  noted  that 
in  any  such  case  the  integral  is 
taken   twice    along   each    of   the 

^-X  cross-cuts,  once  in  either  direction. 

This  portion  of  the  integral  van- 
^°'      ■  ishes  in  accordance  with  the  second 

property  of  line-integrals  already  stated.  We  may  then  say  that 
the  integral  taken  over  the  contour  of  a  multiply  connected  region, 
that  is,  the  sum  of  the  integrals  taken  over  the  several  closed  curves 
constituting  the  boundary,  with  the  proper  signs  attached,  is  the 
same  as  the  integral  taken  over  the  boundary  of  the  simply 
connected  region  formed  by  inserting  cross-cuts  in  the  given 
region.* 

16.  Green's  theorem.  One  of  the  important  theorems  associ- 
ated with  line-integrals  gives,  under  certain  conditions,  a  relation 
between  such  integrals  and  ordinary  double  integrals.  This  theorem, 
known  as  Green's  theorem,  may  be  stated  for  functions  of  two  real 
variables  as  follows : 

Theorem  I.     In  a  given  finite  region  S  let  C  he  the  complete  boundary 

of  any  portion  of  the  plane  siich  that  C  lies  within  S  and  incloses  only 

points  of  S.    If  in  the  given  region  P{x,  y)  and  Q{x,  y)  are  continuoits 

real  functions  of  x  and  y  together,  having  the  continuous  partial  deriva- 

..      dQ    dP  ^, 
fives  -r—>  -r-,  then 
dx     dy 

where  the  double  integral  is  to  be  taken  over  the  region  bounded  by  C. 

*  Cf.  Osgood,  Lehrbuch  der  Funktionentheorie,  2d  Ed.,  Vol.  I,  Chap.  IV,  Art.  4, 
Chap.  V,  Art.  7. 


Abt.  16.] 


GREEN'S  THEOREM 


55 


Let  us  consider  the  double  integral 


If 


dy 


dxdy, 


taken  over  the  region  bounded  by  the  curve  C 
sider  this  curve  to  be  a  single 
closed  curve  such  that  any  parallel 
to  the  F-axis  cuts  it  in  at  most  two 
points  as  indicated  in  Fig.  21.  Let 
A  and  B  be  the  points  of  C  at  which 
X  has  a  minimum  and  a  maxi- 
mum, respectively.  A  parallel  to 
the  F-axis  lying  between  Aa  and 
Bh  cuts  the  curve  C  in  two  points 
whose  ordinates  may  be  denoted 
by  Vh  2/2,  respectively. 

,dP 


We  shall  first  con- 


Because  of  the  continuity  of  -r- , 

dy' 


^X 


Fig.  21. 
the  following  integrals  exist,  and  we  may  write  * 


n%-^y=j>j! 


dP 

dy 


dy. 


(1) 


Performing  the  integration  in  the  second  member  with  respect  to  y, 
we  get  from  this  equation  the  following  relation: 

J  J  ^dxdy  =  £  \P{x,  y,)  -  P(x,  y{)  |  dx.  (2) 

However,  the  two  integrals 

J    Fix,  yi)  dx,        J   P(x,  yi)  dx 

are  line-integrals  taken  along  the  paths  Ay^B,  AyiB,  respectively. 
Hence,  instead  of  the  integral  in  the  second  member  of  (2)  we  may 
write  a  single  integral  taken  in  a  negative  direction  around  the  curve 
C,  or  what  is  the  same,  we  may  write  the  negative  of  that  integral 
taken  in  a  positive  direction  around  C,  and  thus  have 


ff%^'y=-l'' 


dx. 


(3) 


*  See  Hobson,  Theory  of  Functions  of  a  Real  Variable,  Arts.  314  and  315. 


56 


DIFFERENTIATION  AND  INTEGRATION 


[Chap.  III. 


In  a  similar  manner,  we  can  deduce  the  relation 
By  subtracting  (3)  from  (4)  we  have 

Xp<ix  +  ed,  =  //(g-|)dxdy, 

as  the  theorem  requires. 

To  extend  (3)  and  (4)  to  any  finite  region  bounded  by  an  ordinary 
curve  C,  all  that  is  needed  is  to  divide  the  region  into  subregions, 
each  of  which  satisfies  the  condition  that  a  line  parallel  to  the  F-axis, 
in  case  of  (3),  or  to  the  X-axis,  in  case  of  (4),  cuts  the  boundary 
curve  in  not  more  than  two  points,  or  in  a  segment  as  p,  q,  Fig.  22. 

This  is  always  possible  as  indicated 
in  Fig.  22,  since  the  given  contour 
by  hypothesis  has  only  a  finite 
number  of  maxima  and  minima 
with  respect  to  x  and  with  respect 
to  y.  If  the  boundary  C  consists 
of  several  closed  curves,  the  region 
is  multiply  connected.  By .  the 
proper  introduction  of  cross-cuts 
this  region  may  be  made  simply 
connected,  and  the  foregoing  argu- 
-^-X  ment  then  holds.  As  we  have  seen, 
however,  the  value  of  the  integral 
along  C  becomes  in  this  case  the 
sum  of  the  integrals  taken  in  the  proper  direction  along  the  several 
closed  curves  composing  C. 

We  shall  now  consider  the  following  theorem,  which  is  important 
in  subsequent  discussions. 

Theorem  II.  In  a  given  finite  region  S  let  C  be  the  complete  bound- 
ary of  any  portion  of  the  plane  such  that  C  lies  within  S  and  incloses 
only  points  of  S.  If  in  the  given  region  P{x,  y),  Q(x,  y)  are  continuous 
real  functions  of  x  and  y  together,  having  the  continuous  partial  deriva- 


O 


Fig.  22. 


lives 


dQ   dP 

dx     dy 


,  then  the  necessary  and  sufficient  condition  that  the  integral 


f 


Pdx  +  Qdy 


(5) 


Akt.  16.]  GREEN'S  THEOREM  57 

vanishes  for  every  such  curve  C  is  that 

dx       dy  ^^ 

for  all  points  of  S. 

That  this  condition  is  necessary  may  be  established  as  follows. 
We  have  given  the  condition  that  the  line-integral  (5)  vanishes  to 
show  that  the  condition  (6)  follows  as  a  consequence.     The  function 

■^ ^  is  continuous  in  S.     If  this  function  is  not  identically  zero 

dx       dy  -^ 

for  all  values  of  x,  ^  in  S,  then  it  is  possible  to  find  a  subregion  R 
sufficiently  small  such  that  —  —  —  is  of  the  same  sigp  for  all  values 
of  X,  y  in  R.     From  Green's  theorem,  we  have 

fpa.  +  Qay  =  ff(^£^'£)^ay.  (7) 

rid  r)P 

If  in  R  the  function ^  is  always  of  the  same  sign  for  all  values 

of  X,  y,  then  the  double  integral  in  the  second  member  of  (7)  can  not 

vanish  and  consequently  the  line-integral    /  Pdx  +  Qdy  can  not 

equal  zero  when  taken  around  the  contour  of  R.  Hence,  in  order 
that  the  integral  (5)  taken  around  every  complete  boundary  C  in  S 
shall  vanish,  the  condition  (6)  must  be  satisfied  identically  for  all 
values  of  x,  y  in  S. 

That  the  condition  stated  in  the  theorem  is  also  sufficient  follows 
at  once  from  equation  (7) ;  for,  if  (6)  holds  for  all  values  of  x,  y  in  S, 
then  the  integral  (5)  taken  along  any  complete  boundary  C  in  S 
vanishes  as  the  theorem  requires. 

We  are  now  in  position  to  establish  the  following  proposition. 

Theorem  III.  In  a  given  finite  simply  connected  region  S,  let  L 
be  any  ordinary  curve  joining  two  points  of  S  and  lying  within  S.  If 
in  the  given  region  P{x,  y),  Q{x,  y)  are  continuous  real  functions  with 

SO 

respect  to  x  and  y  together,  having  the  continuous  partial  derivatives  —  , 

f)P 

-— -,  then  the  necessary  and  sufident  condition  that  the  line-dntegral 
dy 

j   P  dx-\-  Q  dy  is  independent  of  the  path  L  is  that 

dQ^dP 
dx       dy 
for  all  points  of  S. 


58 


DIFFERENTIATION  AND  INTEGRATION 


[Chap.  III. 


This  theorem  follows  directly  from  the  conclusions  of  Theorem  II. 

Let  (xo,  i/o),  {xi,  yO  be  any  two  points  in  the  region  S  (Fig.  23). 

Let  Li,  Lj  be  any  two  non-intersecting  ordinary  curves  joining  these 

points  and  lying  wholly  in  S.  The 
lines  Li,  L2  taken  together  con- 
stitute a  closed  curve  C  lying  in  S; 

and  if  r—  =  -r-  for 
dx 


dy 


Fig.  23. 


all  points  in 
S,  then  by  Theorem  II  the  integral 
I    P  dx  -\-  Qdy  is  zero.      Hence 
the  two  integrals 

f  Pdx  +  Qdy,      fpdx  +  Qdy 


must  be  equal,  both  integrals  being  taken  in  a  positive  direction  from 
{xq,  yo)  to  {xiy  yi).  In  other  words,  the  line-integral  is  independent 
of  the  path.  Conversely,  if  these  two  integrals  are  equal,  the  inte- 
gral around  the  closed  curve  C  vanishes;  but  C  is  any  ordinary  closed 
curve  in  S  and  hence  by  Theorem  II,  we  have 

dP  ^dQ 

dy       dx' 
as  the  theorem  requires. 

To  apply  the  foregoing  theorem  to  a  multiply  connected  region,  it 
is  necessary  first  to  make  it  simply  connected  by  the  introduction  of 
the  proper  cross-cuts. 

The   integral    I       P  dx  -\-Q  dy  taken   along   an   arbitrary   path 

starting  from  a  fixed  point  (xq,  yo)  and  having  a  variable  upper  limit 
is,  under  the  conditions  set  forth  in  Theorem  III,  a  function  of  x  and 
y.    Moreover,  we  have  the  following  theorem. 

Theorem  IV.  In  a  given  finite  simply  connected  region  S,  let  (xo,  yo) 
be  any  fixed  point  and  (x,  y)  a  variable  point.  If  in  the  given  region  the 
functions  P{x,  y),  Q{x,  y)  are  continuous  real  functions  in  both  x  and  y 

BQ    aP 


having  the  cordinuous  partial  derivatives  — , 

tion  -r-  =  -T— ,  then  the  integral 
dx       dy  ^ 


dy 


,  satisfying  the  condi- 


f" 


Pdx-\-Qdy 


Art.  16.]  GREEN'S  THEOREM  69 

defines  a  function  F{x,  y)  such  that 

dx        '         dy      ^' 

From  what  has  been  said,  we  know  that  the  given  integral  defines 
some  function  of  the  variable  upper  limit.  We  designate  this  func- 
tion by  F{x,  y)  and  proceed  to  show  that  this  function  satisfies  the 
condition  of  the  theorem.  Let  {xi,  yi)  be  any  point  of  S  and  let 
(xi  +  Ax,  yi)  be  a  second  point  of  *S  in  the  neighborhood  of  (xi,  t/i). 
We  have  then 

Pdx  +  Qdy-  /        Pdx  +  Qdy 
Pdx-[-Qdy. 

Since  by  Theorem  III  the  path  of  integration  is  arbitrary,  we  may 
assume  it  to  be  rectilinear,  and  hence  we  can  write 

Pdx  +  Qdy  =    I         Pdx;  (8) 

Xl,  Vi  Jxi 

for,  as  y  does  not  vary  the  value  of  dy  is  zero.  The  resulting  integral 
being  an  ordinary  definite  integral,  we  can  apply  the  first  theorem 
of  the  mean  *  for  such  integrals  and  thus  obtain 

Pdx  =  P(xi  +  dAx, yi)  Ax,        0<e<  1.  (9) 

From  (8)  and  (9)  we  now  obtain 


t/Xl, 


I 


Fix,  +  AX,  y.)  -  Ffa,  y.)  ^  p^^^  _^  ^^_  ^^_  ] 


Ax 
Taking  the  limit  as  Ax  =  0,  we  have 

=  P(xi,yi). 


dFl 


In  a  similar  manner,  we  may  show  that 

T-  =  Q(xi,  yi). 

oyjxuvi 

Since  Xi,  yi  is  any  point  of  S,  we  may  write 

for  all  values  of  x,  y  in  S. 

*  See  Goursat-Hedrick,  Mathematical  Analysis,  p.  151. 


60  DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 

17.  Integral  of  /(«).  We  shall  now  consider  the  integral  of  a 
function  of  a  complex  variable.  Let  J{z)  be  a  continuous  single- 
valued  function  of  z  along  a  given  ordinary  curve  C  joining  the  two 
points  a,  ^.  Between  the  points  a  =  2<,  and  ^  =  z^  insert  n  —  1 
division  points  of  the  curve,  Zi,Z2,...,  z„-i.  Form  the  sum  of  the 
products  f(Sk)  ^kZ,  where  A^z  is  the  difference  Zk  —  Zk-j  and  f(^k)  is 
the  value  of  the  given  function  f{z)  at  some  point  ^k  on  the  curve  C 
between  ^k-^  and  Zk.  The  integral  of  f{z)  along  C  between  the  limits 
a  and  /3  is  defined  as  the  limit  of  this  sum  as  the  number  of  division 
points  between  a  and  /3  is  increased  indefinitely  and  each  difference 
I  Zk  —  Zk-i  I  approaches  zero;  that  is, 

f  f{z)dz  =  L  i;/(f.)A,2.  (1) 

The  existence  of  this  integral  can  be  established  either  directly,  as 
in  the  case  of  real  variables,  or  by  making  it  depend  upon  the  existence 
of  the  line-integrals  already  discussed.  We  shall  choose  the  latter 
method.    For  this  purpose  we  write 

f{z)  =  u{x,  y)  +  iv{x,  y), 

where  u,  v  are  real  functions  of  the  real  variables  x,  y,  and  hence  we 
have  by  putting  ffc  =  ^fc'+  i-qk,' 

Xfi^k)  Akiz)  =Xm)  {zk  -  Zk-i) 

i-l  *=! 

n 

=  ^\u{^h,  rik)(xk  -  Xk-i)  -  V  (^k,  •(]k)(yk  -  yk-i)i 

i=l 
n 

+  i^\v(^k,  ■{ik)(xk  -  Xk-i)  +  w(|*,  'fjk)(yk  -  yk-\)\. 

Since  f{z)  is  continuous  in  z  along  C,  it  follows  from  Theorem  II,  Art. 
13,  that  both  u{x,  y),  v{x,  y)  are  continuous  in  x,  y  together  along  the 
same  curve.  Moreover,  as  A2  =  0,  we  have  Arc  =  0,  At/  =  0. 
Hence,  upon  passing  to  the  limit,  we  obtain 


Jf{z)  dz=    I  (udx  —  V  dp)  +  *  /  (vdx-i-  u  dy); 
c  Jc  Jc 


(2) 


for,  from  the  discussion  of  line-integrals  it  follows  that  the  two  inte- 
grals in  the  second  member  of  this  equation  both  exist  because  of 


Art.  17.]  INTEGRAL  OF  j{z)  61 

the  continuity  of  w(x,  y),  v{x,  y),  and  hence  the  integral  /  f(z)  dz 

must  also  exist. 

In  the  case  of  simple  functions  the  integral  between  two  given 
points  on  a  given  curve  may  be  evaluated  by  direct  application  of  the 
definition  (1). 

Ex.  1.   Evaluate  the  integral    |    dz. 

This  integral  is  independent  of  the  path  over  which  it  is  taken;  for,  by  defi- 
nition, we  have 

n 

Jdz  =     L      ^(Zk  —  Zk—l)   =     L    (Zi  -  Zo  +  Z2-  Zl+  2j  —  22  H 4-  «n  —  2n-l) 
a                n  =  oo  ^^                                  n=oo 
i  =  l 

=     L    (Zn  —  Zo)   =  P  —  a, 
n=ao 

since  zo  =  a,  Zn  =  /3,  no  matter  what  the  intermediate  points  may  be. 

The  geometric  interpretation  (Fig.  24)  of  this  result  will  enable 
the  student  to  understand  more  clearly  the  nature  of  a  definite 
integral  of  a  function  of  a  complex  variable.  Such  an  integral  was 
defined  as  the  limit  of  the  sum^  f{^k)(.Zh  —  Zk-i)',  that  is,  we  consider 
the  limit  of  a  sum  of  prod- 
ucts, each  of  which  is  the 
value  of  the  given  function 
at  some  point  ft  on  the 

curve  multiplied  into  the       ///^^        __— ' ^^'^*' 

directed    chord    Zk  —  Zk-\. 
In  this  particular  case,  the        *  Pt     94 

value  of  /(^a)    is  always 

unity.  Adding  Zi  —  Zo  to  Z2  —  Zi,  we  have  geometrically  the  directed 
chord  zi  —  zq.  Adding  to  this  result  the  directed  chordzs  —  ^2,  we 
have  the  directed  chord  Zs  —  Zo,  etc.  Finally,  we  have  Zn  —  zo, 
which  is  identically  /3  —  a  as  we  have  seen.  This  result  is  very  dif- 
ferent from  taking  the  integral  /    \dz\.     In  this  case,  we  add  merely 

n 

the  chords  without  reference  to  direction;  that  is,  we  have  L  ^  |  AkZ  |. 

In  the  limit  we  should  have  in  this  dkse  not  /3  —  a  but  the  length  L 
of  the  path  *  of  integration  from  a  to  jS. 

*  See  Townsend  and  Goodenough,  First  Course  in  Calculus,  Art.  88. 


62  DIFFERENTIATION   AND   INTEGRATION         (Chap.  III. 

zdz.  I    ^ 

This  integral  is  independent  of  the  path  over  which  it  is  taken;  for,  ^hylimit 
defining  the  int^ral  exists  when  ft  is  any  point  in  the  interval  zt-i,  ^  zL  We 
may,  therefore,  select  for  ft  any  convenient  point  in  this  interval.  If  we  ta^  it  to 
be  Zk  or  Zk-\,  we  have  respectively 

2dz=  L   ^Zk{zk  —  Zk-i),     or     \    zdz=  L  ^  Zk-i{zk  —  Zk-i) . 
Hence,  we  have  by  taking  one-half  of  the  sum  of  these  two  results 


X 


L   ^{zk^-Zk-i^) 

*  11—00  i._i 


11-00  j_| 

zdz  = 


=   L 


2 

(2x2  -   V  +  32^  -  2l'  +  23^   -   22^+     •    •    •     +  Zn^  -  Zn-l^) 


L    (2»''  -  20^)         (/S^^  -  «^) 

11  —  00 


no  matter  what  the  path  is,  since  as  in  Ex.  1,  zo  =  «,  2„  =  jS, 

In  both  of  these  examples  the  result  obtained  is  the  same  as  that 
obtained  by  substituting  the  limits  of  integration  in  a  function  of 
which  the  integrand  is  the  derivative  and  taking  the  difference. 

The  definite  integral  of  a  function  of  a  complex  variable  has  been 
defined  as  a  limit.  From  this  definition  and  the  laws  of  operation 
with  limits,  the  general  properties  of  such  integrals  can  be  deduced ; 
or,  they  may  be  shown  to  hold  as  a  consequence  of  the  corresponding 
properties  of  line-integrals  in  the  calculus  of  real  variables.  The  proof 
in  many  cases  is  so  evident  that  it  is  left  to  the  reader  to  supply. 
If  a,  /3  be  two  points  on  the  path  of  integration  C,  we  then  have 
among  other  properties: 

1.  J^f{z)dz  =  -jy{z)dz, 

2.  /    cf{z)  dz  =  c  I   f{z)  dz. 

3.  P[/(z)  ±  <!>(«)]  dz  =  rf{z)  dz  ±  r^{z)  dz. 

i/a  Jo,  Ja 

This  last  property  can  be  readily  extended  to  the  case  involving 
any  finite  number  of  functions.     It  can  not,  however,  be  extended 


Art.  17.]  INTEGRAL  OF  j{z)  63 

to  the  case  involving  an  infinite  number  of  functions  without  intro- 
ducing some  condition  as  to  the  character  of  the  convergence  of  the 
series  thus  introduced. 


4.     /    f{z)  dz  =  C\z)  dz-^  f  f{z)  dz, 


where  Zi  lies  upon  the  path  of  integration  C  connecting  a  and  /3. 

This  property  can  be  extended  to  the  case  where  the  path  of  inte- 
gration is  broken  up  into  a  finite  number  of  parts  by  inserting 
between  a  and  j8,  n  —  1  points  Zi,  Z2,  .  .  .  ,  Zn-i  on  the  path  of 
integration.  If  the  ordinary  curve  constituting  the  path  of  inte- 
gration C  is  composed  of  a  finite  number  of  connected  lines  Ci, 
Cz,  .  .  .  ,  C„,  we  write 

ff(z)dz=    f  f{z)dz+   f  f{z)dz+  +   f  J{z)dz. 

This  relation  enables  us  to  extend  the  definition  of  an  integral  to 
include  integrals  taken  over  the  contour  of  a  multiply  connected 
region.  As  in  line-integrals  of  functions  of  real  variables,  the  inte- 
gral is  said  to  be  taken  in  a  positive  direction  with  respect  to  the 
region  bounded  when  it  is  taken  in  a  positive  direction  with  respect 
to  this  region  about  each  closed  curve  constituting  a  portion  of  the 
boundary.  The  integral  over  the  complete  boundary  is  unaffected 
by  the  introduction  of  the  cross-cuts  necessary  to  make  the  given 
region  simply  connected. 

5.  \rf{z)dz\^   f^\f{z) 
I  Jo.  I       J  a 

We  have 

X/(r.)  A.J  ^  i;  I  /(f .)  A.2 1  =  X I  /(f  ^)  1  •  I  ^^-^  I' 

i=l  I         i=l  i=l 

since  the  absolute  value  of  a  sum  is  less  than  or  at  most  equal  to  the 
sum  of  the  absolute  values  of  the  terms,  and  the  absolute  value  of  a 
product  is  always  equal  to  the  product  of  the  absolute  values  of  the 
factors.  Passing  to  the  limit  as  Az  approaches  zero,  we  have  the  re- 
quired relation. 
From  Ex.  1,  we  have 

6.  J^\dz\=L, 

where  L  is  the  length  of  the  path  of  integration. 


dz 


64  DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 


7.  IX^^^^)''^ 


=     M'    Ly 


where  M  is  the  maximum  value  of  \  f(z)  \  along  the  path  of  integration 
and  L  is  the  length  of  that  path. 

The  result  stated  in  this  theorem  follows  at  once  from  (5)  and  (6) ; 
for,  we  have  upon  replacing  |  /(«)  |  by  its  maximum  value 


P/(2)  dz    ^M  f 

*J  a  *J  a 


^  dz\  =  M'L. 


18.  Change  from  complex  to  real  variable.  We  can  readily  de- 
duce the  law  for  the  change  of  the  independent  variable  in  a  definite 
integral  of  a  complex  variable.  We  shall  first  consider  the  change 
from  a  complex  variable  to  a  real  variable.  We  have  from  equation 
(2),  Art.  17, 

Jf{z)  dz  =    j     udx  —  vdy-\-ij     vdx-\-udy,  (1) 

AB  Jab  Jab 

where  f{z)  =  u(x,  y)  +  ivix,  y). 

By  aid  of  equation  (11),  Art.  15,  we  may  express  the  two  integrals  in 
the  second  member  of  (1)  in  terms  of  a  parameter  t  and  thus  obtain 

/     udx-vdy  =    /'"{w  .^i'(0  -  v  -^z'CO  1  dt,  (2) 

Jab  Jte 

f   vdx  +  udy=    ]     \v'^i'{t)  +  m -^/(O \  dt,  (3) 

*Jab  Jut 

where  as  in  Art.  15 

a:=^i(0,         2/ =^2(0,         to  =  t^tn 

is  the  parametric  equations  of  the  path  of  integration.    By  combin- 
ing (2)  and  (3),  we  have  from  (1), 

/    f{z)dz=  J     \u'^i'{t)-V'^2'{t)ldt-\-i  r''lv'^i'{t)-\-U'^2\t)]dt 

V  AB  tJtt  fj i^ 

=  /'"  (u  +  iv)  \^i'(t)  +  i^,'{t)  \  dt.  (4) 

Remembering  that  D-O' 

Dtz  =  D,[<iri(t)  +  1^2(0]  =  ^i'(0  +  i^z'it), 


Art.  18.]  CHANGE  OF  VARIABLE  65 

and  putting 

j{z)  =  u{x,  y)  +  iv{x,  y) 

=  F{t), 
we  may  write  the  above  result  in  the  following  compact  form: 

f  f{z)dz=    r''Fit)DtzdL  (5) 

We  thus  obtain  precisely  the  same  rule  for  change  of  variable  in  the 

integral  /     f{z)  cfe  as  is  ordinarily  formulated  for  the  definite  integral 
Jab 

of  a  function  of  a  real  variable;  namely,  substitute  in  f{z)  dz 

z  =  Mt)  +  ^'^2(0,         dz  =  {^i'(0  +  i^2\t)  \  dt, 

with  a  corresponding  change  in  the  limits  and  the  path  connecting 
them.  Equation  (4),  or  its  equivalent  equation  (5),  gives  a  means 
by  which  the  calculation  of  an  integral  of  a  function  of  a  complex 
variable  may  be  made  to  depend  upon  the  evaluation  of  at  most  four 
ordinary  definite  integrals  of  functions  of  a  real  variable. 

Ex.  1.     Evaluate  the  integral  I    ,  where  C  is  a  circle  of  radius  p  about  a. 

J  cz  —  a 

Put  z  —  a  =  p(cos  6  +i  sin  d), 

whence  Dgz  =  p(— sin  0  +i  cos  6) 

=  ip{cos  6  +isind). 

As  z  describes  the  circle  C,  9  passes  from  0  to  2  tt.     We  have  then 

=    I       -7 — -; — 7—. — rr  •  tp(cos  B  -\- %  sin  d)  dd 

cz  —  a       Jo     p(cos  0  +  I  sin  6) 

=   j      idd 

Jo 

=  2jri. 

J*        dz 
r^,  where  C  is  the  same  as  in  Ex. 
c  \^  ~  '^) 
1,  and  n  is  an  integer  different  from  one. 
As  in  the  preceding  example,  put 

z  —  a  =  p(cos  0  +  i  sin  0). 
We  then  have 

Jdz       _    r^it  ipjcos  0  +  i  sin  6)  dd 
Q  (z  -  a)"  ~  Jo       p"(cos0  +  isin0)" 

=  -^  r  '  f  cos  (n  -  1)  e  -  t  sin  (n  —  1)  e  I  rffl 
"       «/o 


i 

-n— 1 


]   )      COS  {n  —  \)  Odd  —  i   \      s,m{n  —  I)  6 de\. 

Jo  -^0 


66  DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 

But  from  the  calculus  of  real  variables,  we  have  for  n  7»^  1, 

r"co8(n- l)0dd  =  0,         J'^smCn- l)ffd»  =  0. 

Hence,  we  obtain  the  resiilt 


X 


^       =0,        n  ^  1. 


c  («  -  «) 

We  shall  consider  later  (Art.  22)  the  case  of  change  of  variable  from 
one  complex  variable  to  another. 

19.  Cauchy-Goursat  theorem.  The  properties  of  definite  in- 
tegrals considered  in  Art.  17  depend  upon  the  condition  that  f{z) 
is  a  continuous  function.  Consequently,  they  hold  where  the  given 
function  is  holomorphic,  since  such  a  function  is  necessarily  contin- 
uous. 

We  shall  now  consider  some  of  the  special  properties  of  integrals 
of  functions  which  are  holomorphic  in  a  given  region.  The  most 
important  and  fundamental  of  these  properties  is  stated  in  a  theorem 
due  originally  to  Cauchy.  The  proof  of  this  theorem  may  be  made 
to  depend  upon  Green's  theorem  but  the  results  obtained  can  then  be 
said  to  hold  only  under  the  initial  restrictions  assumed  in  the  demon- 
stration of  that  theorem.  Goursat  has  shown  *  that  the  condition 
that  the  derived  function  f{z)  is  continuous  is  not  necessary  for  the 
demonstration.  In  order  to  establish  the  theorem  without  assuming 
this  condition,  the  following  lemma  will  be  of  use. 

Lemma.  Given  a  region  T  in  which  f(z)  is  holomorphic.  Let  the 
ordinary  closed  curve  C,  lying  wholly  in  T  and  containing  only  points 
of  T,  be  the  complete  boundary  of  a  region  T'.  It  is  always  possible  to 
divide  the  region  T'  into  a  finite  number  of  squares  Si  and  partial  squares 
Ri  such  that  vnthin  or  upon  the  boundary  of  each  of  these  subregions  there 
exists  a  point  Zi  such  that  as  z  describes  the  boundary  of  the  subregion 
we  have 

f(z)-f(Zi) 


Z  —  Zi 


<  e,    ■  (1) 


where  e  is  a  previously  assigned  arbitrarily  small  positive  number. 

The  boundary  curve  C  (Fig.  25)  is  by  hypothesis  an  ordinary  curve 
and  hence  has  but  a  finite  number  of  maxima  and  minima  with  respect 
to  X  and  with  respect  to  y.    Consequently,  by  drawing  lines  parallel  to 

*  See  Trans.  Amer.  Math.  Soc,  Vol.  I,  pp.  14-16;  Moore,  Ibid.,  pp.  499-506; 
Pringsheim,  Trans.  Amer.  Math.  Soc.,  Vol.  II,  pp.  413-421. 


Art.  19.] 


CAUCHY-GOURSAT  THEOREM 


67 


-^ 

A 

\ 

V     J- 

/ 

Yi 

/I 

/ 

/ 

+ 

/ 

/ 

/ 

1 

V 

*  / 

/ 

\ 

^ 

-^^ 

/ 

the  two  axes  of  coordinates  we  can  cover  the  given  region  T'  with  a 
system  of  congruent  squares,  such  that  the  perimeter  of  no  square  is 
cut  by  the  contour  C  in  more  than  two  points.  Let  c  denote  the 
length  of  a  side  of  these  squares  and  let  A  denote  the  combined  area 
of  these  squares.  By  this  means 
the  given  region  T'  is  subdivided 
into  smaller  regions.  Some  of  these 
regions  are  complete  squares  and 
others  are  partial  squares  along 
the  contour  C,  bounded  in  part  by 
straight  line  segments  and  in  part 
by  arcs  of  C  There  may  or  may 
not  exist  within  or  upon  the  bound- 
ary of  each  of  these  subregions  a 
point  Zi  satisfying  the  conditions  of 
the  lemma. 

If  in  any  subregion  such  a 
point  does  not  exist,  we  divide  the 
corresponding  square  into  four 
equal  squares  by  drawing  lines  par- 
allel to  the  two  axes  of  coordinates.  Those  squares  or  partial  squares 
satisfying  the  condition  given  in  (1)  are  left  unchanged.  Moreover, 
we  consider  only  those  new  squares  and  partial  squares  that  lie  in 
the  region  T' .  If  any  of  these  new  squares  or  parts  of  squares  satisfy 
the  required  condition,  they  are  not  subjected  to  further  subdivision. 
The  process  of  subdivision  is,  however,  continued  with  the  rest. 
In  this  way  either  there  is  ultimately  obtained  a  finite  number  of 
subregions  of  the  desired  character,  or  there  exists  at  least  one  in- 
finite sequence  of  squares  each  lying  within  the  preceding,  such  that 
these  squares  or,  in  case  the  squares  contain  points  not  in  T' ,  the  cor- 
responding partial  squares  in  no  case  satisfy  the  condition  set  forth  in 
(1).  The  sides  of  the  squares  of  this  infinite  sequence  approach  zero 
as  a  limit,  and  the  sequence  satisfies  the  conditions  of  Theorem  IV 
of  Art.  12.  Consequently,  such  a  sequence  defines  a  definite  limiting 
point  a. 

The  point  a  is  a  regular  point  of  the  function  /(z) ;  hence  the  deriv- 
ative /'(a)  exists  and  we  have 


I 

Fig.  25. 


L^M^l^M.  =;.(„), 


(2) 


68  DIFFERENTIATION  AND  INTEGRATION         [Chap,  III. 

where  z  —  a  is  the  increment  of  z.    This  relation  can  be  written  in 
the  form 

f(z)-fM 


-/'(«) 


<  6,  (3) 


z  —  a 

which  holds  for  all  values  of  z  in  T'  such  that  \  z  —  a\  <  8.  Draw 
about  the  point  a  a  circle  of  radius  5.  From  some  point  on  in  the 
sequence  of  regions  defining  the  point  a,  all  of  the  regions  lie  within 
this  circle,  and  consequently  if  a  is  taken  as  the  point  Zi  the  values 
of  z  upon  the  perimeter  of  any  one  of  these  regions  are  such  that  (1) 
is  satisfied.  This  conclusion  contradicts  the  assumption  that  none  of 
the  subregions  of  the  sequence  satisfies  the  required  condition.  From 
this  contradiction  the  given  proposition  follows. 

By  aid  of  this  lemma,  we  may  now  demonstrate  the  Cauchy- 
Goursat^th^g rpjn ^  which  may  be  stated  as  follows: 

Theorem  I.  Let  f{z)  he  holomorphic  in  a  given  finite  region  S  and 
let  C  be  the  complete  boundary  of  any  portion  S'  of  S  such  that  C  lies 
wholly  in  S  and  incloses  only  points  of  S;  then 


£ 


f{z)  dz  =  0. 
C-'    ' 

The  boundary  C  may  consist  of  a  single  closed  ordinary  curve  or  a 
combination  of  such  curves.  We  shall  first  consider  the  case  where 
Cis  a  single  closed  ordinary  curve  and  the  inclosed  region  &'  is  simply 
connected.  Let  &'  be  divided  into  squares  and  partial  squares  satis- 
fying condition  (1)  of  the  foregoing  lemma.  Let  n  denote  the  number 
of  squares  S,-  and  m  the  number  of  partial  squares  i2,.  If  the  integral 
is  taken  in  a  positive  direction  around  the  perimeter  of  the  various 
subregions  S,,  jR„  it  will  be  seen  that  each  side  of  these  regions  that  is 
not  a  portion  of  C  is  taken  twice  as  a  path  of  integration,  the  two  in- 
tegrals being  taken  however  in  opposite  direction.  Considering  the 
sum  of  the  integrals  about  the  perimeters  of  all  of  the  regions  <S„  i2„ 
we  may  therefore  write  by  aid  of  4,  Art.  17, 

Jr»  ^         f*  vet         r% 

f(z)  dz  =  x    m  dz  +  X    /(^)  ^^'  (4) 

where  yi,  \i  denote  the  boundaries  of  <S„  Ri,  respectively. 

From  the  lemma,  we  have  within  or  upon  the  boundary  of  each 
Si,  Ri  a  point  z*  such  that 

f(z)-f(zi) 


z.      -/'(^^) 


<  e.  (5) 


Art.  19.]  CAUCHY-GOURSAT  THEOREM  69 

This  relation  can  be  written  in  the  form 

/(2)  -  f{z.)  =  {z-  2i)  S'iz.)  +  -miz  -  Zi),       ,  (6) 

where  iQi  is  a  function  of  z  such  that     — —  ol  M-^'-^^^^^  '^'^  > 

hil<6,  (7) 

when  z  varies  along  the  contour  of  Si  or  Ri.  We  shall  now  consider 
the  integral  around  the  perimeter  of  one  of  the  squares  Si.  We  have 
from  (6),  since  Zi  is  a  constant  for  this  integration, 

f  f(z)dz=[f(zi)-Zif'izi)]  f  dz+f{Zi)  f  zdz-\-  fr,i{z-Zi)dz.    (8) 

From  Exs.  1,  2,  Art.  17,  we  know  that 

dz  =  ^-a,  I     zdz  =  ^{^^-  a^). 

a  O  a 

In  the  particular  case  under  consideration,  as  the  path  of  integra- 
tion is  a  closed  curve,  a  and  j8  are  the  same  point,  and  hence  both  of 
these  integrals  vanish.     From  equation  (8)  we  then  have 


U/(z)  dz\  =  \  I  TQf(z  -  Zi)  dz 


(9) 


Let  the  length  of  one  side  of  the  square  Si  be  d.     The  diagonal 
of  the  square  is  then  d  V'2.     Hence,  we  have 

\z  -  Zi\  =  Ci  V2. 

Making  use  of  this  relation  and  of  that  given  in  (7),  we  may  now 
write  by  7,  Art.  17, 

I  ffiz)dz   <eCiV2  f  \dz\=eCiV2'^Ci  =  e4:V2Ai,  (10) 

where  A  i  denotes  the  area  of  Si. 

Consider  now  the  integral  taken  around  one  of  the  partial  squares 
Ri.    We  have 

f  f{z)dz={f{z.)-Zif{z.)]  f  dz+f(zi)  f  zdz+  f  miz-Zi)  dz.  (U) 

As  before  the  first  two  integrals  in  the  second  member  of  this  equation 
vanish  and  we  have 


/  /(z)  dz\  =  \  I   rii{z  -  Zi) 

J\i  I         I  «/Xi 


dz  •  (12) 


70  DIFFERENTIATION  AND  INTEGRATION  [Chap.  in. 

We  may  denote  by  c,  the  length  of  a  side  of  the  square  of  which  Ri 
is  a  portion,  Ri  being  that  portion  of  the  square  cut  off  by  curve  C 
and  lying  in  S'.  Let  Z,  be  the  length  of  that  arc  of  C  which  forms ^ 
portion  of  the  boundary-  of  Ri.  We  have  then  \z  —  Zi\  =  d V2. 
From  (12),  we  have 

\ff{z)dz  <  eCiV2f  \dz\<eCiV2i4:Ci-{-li)^€V2{4.Bi-\-cli),   (13) 

where  B,  denotes  the  area  of  the  square  of  which  Ri  is  a  part,  and  c 
is  the  length  of  one  side  of  the  largest  square  that  comes  into  consider- 
ation in  the  subdivision  of  S'. 

Replacing  each  term  of  the  sums  in  (4)  by  its  absolute  value,  we 
have,  by  use  of  (10)  and  (13), 

I  ffiz)dz    <6V2{y4^i  +  2(4Bi  +  cZ0[ 

^eV2]4A-hcLl  (14) 

where  L  denotes  the  length  of  the  curve  C,  and  A  denotes,  as  in  the 
discussion  of  the  lemma,  the  combined  area  of  the  system  of  congru- 
ent squares  with  which  the  region  S'  was  originally  covered.  The 
expression  included  in  the  braces  is  therefore  a  constant,  and  as  c  is 
arbitrarily  small  the  product  is  arbitrarily  small.     As  the  absolute 

value  of  the  integral    i  /(«)  dz  is  shown  to  be  less  than  an  arbitrarily 

small  number,  it  follows  that 

fj{z)dz  =  0, 

as  required  by  the  theorem. 

The  above  theorem  is  now  established  for  the  case  where  the 
region  S',  bounded  by  C,  is  a  simply  connected  region.  The  proof 
may  readily  be  extended  to  the  case  where  S'  is  multiply  connected. 
By  introducing  the  necessary  cross-cuts,  S'  becomes  simply  con- 
nected and  the  foregoing  proof  applies.  However,  in  taking  the  in- 
tegral around  the  boundary,  including  the  cross-cuts,  these  cross-cuts 
are  traversed  twice,  once  in  each  direction.  As  pointed  out  in  Art. 
17,  this  portion  of  the  integral  vanishes,  and  we  have  the  theorem 
applying  to  the  complete  boundary  C  of  the  multiply  connected 
region  aS'. 


Art.  19.] 


CAUCHY-GOURSAT  THEOREM 


71 


In  the  statement  of  Theorem  I,  it  is  assumed  that/(2)  is  holomorphic 
in  the  region  S'  including  its  boundary  C.  We  shall  now  show  that 
it  is  sufficient  that  f(z)  is  holomorphic  within  the  region  bounded  by 
C  and  converges  uniformly  to  its  values  along  C.  As  we  have  already 
seen  (Art.  13),  such  a  condition  is  equivalent  to  saying  that  the  values 
of  f{z)  along  C  are  continuous  with  the  values  of  the  function  within 
the  region  bounded  by  C.  Moreover,  uniform  convergence,  together 
with  continuity  of  f{z)  within  the  region  bounded,  enables  us  to  say 
that  f{z)  changes  continuously  as  z  varies  continuously  along  C.  In 
certain  discussions,  the  conditions  given  in  the  following  theorem  will 
be  more  convenient  than  those  of  Theorem  I. 

Theorem  II.  Iff{z)  is  holomorphic  within  a  finite  region  S  bounded 
by  an  ordinary  curve  C  and  if  it  converges  uniformly  to  its  values  along 
C,  then 


L 


Fig.  26. 


f{z)  dz  =  0. 


Since  /(z)  converges  uniformly 
along  C,  it  follows  from  the  dis- 
cussion of  uniform  convergence 
(Art.  13)  that  about  each  point  z  of 
C  there  may  be  drawn  a  partial 
circle  of  radius  p,  which  is  the  inde- 
pendent of  \  such  that  for  all  values  of  t  within  this  partial  circle 
we  have 

\fit)-f{z)\<e,  \t-z\<p. 

Since  this  condition  holds  simultaneously  for  all  values  of  z  along  C, 
there  exists  a  closed  curve  C  such  that  as  t  traverses  C  we  have 

t  =  Zi>  +  d{z-Zo),\    0  <  0  <  1, 

where  Zo  is  a  fixed  point  interior  to  the  region  bounded  by  C  and  d  is 
a  constant.  The  difference  of  the  integrals  of  the  given  function 
taken  along  the  two  curves  C  and  C  gives* 

ff(z)dz-   f  f(.t)dt=    ff(z)dz-d  f  f[zo-\-diz-Zo)]dz 
Jc  *Jc  Jc  *Jc 

=    f  \f{z)-df[z,  +  d{z-z,)]\dz.       (15) 


*  See  Art.  22  for  method  of  change  of  variable. 


72 


DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 


But  the  int^rand  in  the  last  of  these  integrals  may  be  written  in  the 
form 


f{z)-ef[z-{z-Zo)a 

=  /(2)-/[2-(2 


^)1 

20)  (1  -  0)] +(1  -  e)f[z  -  (z  -  20)  (1  -  e)]. 


Since  (1—6)  can  be  taken  arbitrarily  small,  the  right-hand  member 
of  this  equation  can  be  made  as  small  as  choose;  that  is,  it  can  be 
taken  less  than  an  arbitrarily  small  positive  member  ei.  We  then 
have 

I   ff(z)dz-    fmdt    <ei  f\dz\  =  ei'L, 
\Jc  *Jc'  *Jc 

where  L  is  the  length  of  the  curve  C.    But  as  L  is  finite,  the  product 

€1  •  L  is  arbitrarily  small.     The  integral   /  f{t)  dt  is  zero  by  Theorem 

I.    Hence  we  have 

ffiz)  dz  =  0. 
*Jc 

Theorem  III.    Given  a  finite  simply  connected  region  S  in  which 

the  integral    I  f{z)  dz  vanishes  when  taken  along  any  closed  curve  C 

lying  wholly  urithin  S.  Any  two  paths  of  integration  having  the  same 
extremities  and  lying  wholly  within  S  give  the  same  value  of  the  integral 

Jf{z)dz. 

Let  am/3,  on/S  be  any  two  curves  (Fig.  27)  connecting  the  points 

a,  j8  and  lying  wholly  within  the 
region  S.  We  shall  assume  that 
these  two  curves  do  not  intersect 
each  other.  The  curve  ow/S  fol- 
lowed by  the  curve  /3na  constitute 
a  closed  curve  C.  Then  by  hypo- 
thesis, we  have 


Fig.  27. 


ff(z)dz=   f  J{z)dz  + 
f   f{z)dz  =  0, 


or  by  reversing  the  direction  in  which  the  integral  is  taken  along 
fina,  we  have  by  1,  Art.  17, 


Art.  19.] 


whence 


CAUCHY-GOURSAT  THEOREM 
f    f{z)dz-    f  f(z)dz  =  0; 
f    f{z)dz=    f    f(z)dz, 


7a 


as  required  by  the  theorem. 

It  follows  that,  under  the  conditions  set  forth  in  the  theorem,  the 

value  of  the  integral    I  f(z)  dz  depends  upon  the  limits  between 

which  the  integral  is  taken,  but  not  upon  the  path.  If  one  of  these 
limits  of  integration,  say  /3,  is  replaced  by  the  variable  z,  the  integral 
is  a  function  of  the  upper  limit  and  we  may  write 


J{z)dz  =  F{z). 


Consequently,  we  have  the  following  corollary. 

Corollary  I.     Given  a  finite  simply  connected  region  S  in  which  f(z) 

is  hMomorphio.     The  integral    j  f{z)  dz  taken  along  any  two  paths 

joining  the  same  two  fixed  points  of  S  and  lying  wholly  within  S  has  the 
same  value  for  the  two  paths.  If  one  limit  of  integration  is  a  variable, 
the  integral  is  then  a  function  of  that  limit. 

This  corollary  is  equivalent  to  saying  that  if  f(z)  is  holomorphic 
in  a  simply  connected  region  S  then  a  path  of  integration  between 
two  points  of  S  can  always  be  deformed  into  any  other  path  lying 
wholly  within  S  and  joining  the  same 
two  points  without  affecting  the  value 
of  the  integral. 

If  the  given  region  is  not  simply 
connected,  it  may  be  made  so,  as 
has  been  already  pointed  out,  by 
the  proper  insertion  of  cross-cuts 
(Fig.  28).  Of  course  the  cross-cuts 
then  form  a  part  of  the  boundary  and 
may  not  be  crossed  by  the  path  of 
integration.  In  the  resulting  region 
the  foregoing  proposition  applies,  and  consequently  we  have  the  fol- 
lowing generalization  of  the  foregoing  corollary,  namely: 

Theorem  IV.    Let  S  he  any  region  in  which  f{z)  is  holomorphic 

except  at  most  at  certain  points.     If  a  path  of  integration  of    j  f{z)  dz 


Fig.  28. 


74 


DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 


between  any  two  distinct  points  a,  fi  of  S  is  deformed  into  any  other  path 
between  these  same  points  such  that  it  lies  wholly  in  S,  the  value  of  the 
integral  is  not  affected,  provided  that  in  the  continuous  deformation  of 
the  one  path  into  the  other  no  singular  point  of  f{z)  is  encountered. 

If  the  path  of  integration  is  closed,  that  is  if  a  and  /3  become  coin- 
<cident,  it  follows  that  we  may  replace  any  such  path  Ci  (Fig.  29) 

by  any  other  closed  path  C2, 
provided  that  in  the  region  *S' 
bounded  by  the  two  curves  no 
singularity  oi  f(z)  is  to  be  found; 
'  for,  by  introducing  a  cross-cut  as 
shown  in  the  figure,  the  curve  C2 
together  with  the  cross-cut  may 
be  regarded  as  obtained  by  means 
of  a  continuous  deformation  of  Ci. 
But  as  the  cross-cut  is  traversed 
twice,  once  in  each  direction,  in 
taking  the  integral  along  the  closed  path  from  a  back  to  the  same 
point,  it  may  be  omitted.  For  a  closed  path  of  integration,  we 
have  therefore  the  following  theorem. 

Theorem  V.     In  any  region  S  in  which  f{z)  is  holomorphic  except 

at  certain  points,   a  closed  path  Ci  of  integration  j  f{z)  dz  may  he 

replaced  by  any  other  closed  path  d  lying  either  interior  or  exterior  to 
C\,  but  lying  wholly  within  S,  provided  the  region  bounded  by  the  two 
curves  incloses  no  singular  points  of  f{z)  and  no  points  not  belonging 
ioS. 

Theorem  VI.  If  f(z)  is  holomorphic  in  a  finite  closed  multiply 
connected  region  S  bounded  by  an  exterior  curve  C  and  a  finite  number 
of  inner  curves  Ci,  .  .  .  ,  c„,  then 


Fig.  29. 


ff{z)dz=^  ff{z)dz, 


each  integral  being  taken  in  a  positive  direction  with  respect  to  the  region 
inclosed. 

Connect  each  inner  curve  c*  with  the  exterior  curve  C  by  a  cross- 
cut, thus  making  the  region  simply  connected.  From  Theorem  I, 
the  integral  taken  around  the  complete  boundary,  including  the 
cross-cuts,  is  zero.     However,  it  was  shown  in  Art.  17  that  this  inte- 


Art.  20.] 


CAUCHY'S  INTEGRAL 


75 


gral  is  equal  to  the  integral  taken  over  the  boundary  of  the  multiply- 
connected  region.     We  have  then 


ff{z)dz-\-X   fmdz  =  0. 


(15) 


If  now  the  integrals  along  the  curves  Ck  are  taken  in  a  positive  direc- 
tion with  respect  to  the  regions  interior  to  these  curves  rather  than 
to  the  region  S,  the  direction  in  which  each  integral  is  taken  is 
changed,  and  hence  by  Theorem  I,  Art.  17,  we  have 


f  f(z)dz=j;,  ff(z)dz, 


as  stated  in  the  theorem. 

20.  Cauchy's  integral  formula.  The  following  theorem,  known 
as  Cauchy's  integral  formula,  is  of  fundamental  importance,  as  it 
enables  us  to  express  the  value  of  a  function  of  a  complex  variable 
at  any  inner  point  of  a  finite  closed  region  in  which  it  is  holomor- 
phic,  in  terms  of  an  integral  taken  around  the  boundary. 

Theorem  I.  Given  a  finite  closed  region  S  whose  boundary  C  con- 
sists of  a  finite  number  of  ordinary  curves.  If  f{z)  is  holomorphic 
within  S  and  converges  uniformly 
along  C,  or  if  it  is  also  holomorphic 
for  values  along  C,  then  for  any  inner 
point  a  of  S  we  have 


1    rf(z)dz 


In  accordance  with  the  state- 
ment of  the  theorem,  the  bound- 
ary C  may  consist  of  one  or  more 
closed  curves.  For  example,  in  Fig. 
30  the  complete  boundary  consists  of  the  two  curves  Ci  and  C2. 
About  the  point  a  as  a  center,  draw  a  circle  7  of  radius  p  lying  en- 
tirely within  the  region  S  and  inclosing  only  points  of  S.  Denote 
by  *S'  that  portion  of  the  region  S  lying  outside  the  circle  7.  Then  the 
f(z) 


function 


is  holomorphic  in  the  region  *S',  since  by  hypothesis 


76  DIFFERENTIATION   AND  INTEGRATION         [Chap.  III. 

f{z)  is  holomorphic  in  the  region  S.    From  Theorem  I,  Art.  19,  we 
have  upon  integrating  about  the  contour  of  *S' 


fMdz_^  ffjz)  dz  ^  Q^ 
Jc  Z  —  a      Jy  z  —  a 


or  taking  the  integral  about  7  in  a  positive  direction  with  respect 
to  the  region  inclosed  by  7,  we  have  after  transposing  this  integral 
to  the  second  member  of  the  equation 

n{z)  dz  ^  rf(z)  dz 

Jc  z  —  a     Jy  z  —  a. 

This  result  holds  for  all  values  of  p,  provided  the  circle  lies  entirely 
within  S  and  incloses  only  points  of  S  as  stated  above. 
Because  of  the  continuity  of  f{z)  at  the  point  z  =  a,  we  have 

1/(0)  -/(«)!<  6,          \z-a\^b.  (2) 

For  an  arbitrarily  small  e,  let  z  take  values  upon  the  circle  7  whose 
radius  p  is  not  greater  than  5.     Consider  now  the  integral 

From  the  Ex.  of  Art.  18,  we  have  then 

r_i^  =  2^,  (4) 

JyZ  -  a  '   ^ 

To  evaluate  the  second  integral  in  the  right-hand  member  of  (3)  put 

z  —  a  =  p(cos 6  -\-  isind). 
We  then  obtain 

^  Jy      z  —  a  Jy  z  —  a 

=  r^\f{z)-Ka)\idB.  (5) 

By  5,  Art.  17,  we  have 

I  *Jy  Z  —  <X  I  »/  0 

Since  p,  the  radius  of  the  circle  of  integration,  was  taken  less  than 
or  at  most  equal  to  5,  we  have  by  aid  of  (2) 

£'\f(^)  -  /(«)  \'\de\<  e£'\  dd\=2Te. 


Art.  20.] .  CAUCHY'S  INTEGRAL  77 

From  (6),  we  have 


*J  y 


/W-/W,, 


z  —  a 


<  2  re.  (7) 


The  foregoing  integral  has  the  same  value  if  7  is  any  circle  about  a 
as  a  center,  provided  of  course  that  the  circle  lies  wholly  within  S. 
In  other  words,  this  integral  is  independent  of  the  radius  p  and  hence 
is  independent  of  e.  Since  2  we  is  arbitrarily  small,  we  have  there- 
fore 

■/(^^Md,  =  0.  (8) 


*J  y 


Substituting  the  results  given  in  (4)  and  (8)  in  equation  (3),  we 
have 

7(2)  dz 


%J  y 


z  —  a 
Finally,  we  have  from  equation  (1) 

'f(z)dz 


=  /(a).27rt. 


1 


'c  z  —  a 
or 


=  /(«)•  2  Trt, 


/(«)  =    1     fM^,  (9) 

2TnJcZ  —  a  ^' 

which  is  the  desired  result. 

As  a  consequence  of  the  foregoing  theorem,  it  is  of  importance  to 
observe  that  the  values  of  a  function /(z),  which  is  holomorphic  in  a 
finite  closed  region  *S,  are  fully  determined  for  values  within  S  if 
we  know  its  values  upon  the  contour  of  that  region. 

The  following  theorem  is  of  importance  in  the  further  develop- 
ment of  the  theory  of  analytic  functions.  ^ 

Theorem  II.  If  j{z)  is  holomorphic  in  a  given  finite  region  S,  then 
the  derivative  f{z)  is  a  continuous  function  in  S;  moreover,  f'{z)  is 
itself  holomorphic  in  S. 

Let  Zo  be  any  inner  point  of  S  and  let  C  be  an  ordinary  closed 
curve,  or  combination  of  such  curves,  lying  in  S  and  forming  a  com- 
plete boundary  having  likewise  the  point  Zo  as  an  inner  point.  For 
example,  in  Fig.  30  the  complete  boundary  C  consists  of  the  two 
curves  Ci  and  C2.     From  Theorem  I,  we  have 


^^^^-2ViJct^^' 


where  f  is  a  complex  variable  taken  along  the  contour  C. 


^. 


78 


DIFFERENTIATION  AND   INTEGRATION         (Chap.  III. 


Let  Zo  +  Az  be  any  second  point  in  the  neighborhood  of  Zo,  say 
within  the  circle  y  lying  within  C  and  having  Zo  as  a  center  and  p  as 
a  radius.    We  have  then 


/(2o+Az)-/(zo) 
Az 


J_  C SA 

2TriJc{t  —  Zo 


f(t)dt 


Az)  Az 

^j_  r mdt 

2iriJc{t  —  Zo 


-f 


m  dt 


it 


Az)  {t  -  Zo) 


Zo)  Az 
(10) 


However, 


Fig.  31. 


1 


Az 


(<  -  Zo)  («  -  Zo  -  Az)       (t-  Zo)2  '   (e  -  Zo)2  («  -  Zo  -  Az) 

and  consequently, 

f  m  dt  f  m  dt         f  Azfjt)  dt. 

Jc  it  -  Zo)  (<  -  Zo  -  Az)     Jc  (t  -  ZoY^Jc  {t  -  Zo)2  (t-zo-  Az) 


(11) 


We  can  readily  show  that  the  last  of  these  integrals  has  the  limit 
zero  as  Az  =  0.  To  do  so,  let  r  be  the  lower  limit  of  the  distance  of 
any  point  within  y  from  a  point  on  C.    We  have  then 

|<-Zo-Az|>r,         \t-Zo\>r. 
By  use  of  7,  Art.  17,  we  may  now  write 


IX 


Az/(0  dt 


c(t-Zi,y{t-Zo-  Az) 


ML 


Az 


where  M  is  the  maximum  value  of  ]  f(t)  \  along  C  and  L  is  the  length 
of  the  curve  C.  Hence  as  Az  approaches  zero,  we  have  zero  as  the 
limit  of  this  integral. 


Art.  20.1  CAUCHY'S  INTEGRAL  79 

Consequently,  passing  to  the  limit  as  A2  =  0,  we  have  from  (11) 

Az=o  Jc{t  —  Zo){t  —  Zo  —  Az)       Jc  {t  —  ZaY' 
Hence,  from  (10)  we  get 

.      ^    /(go  +  A2)-/(2o)  ^  J_     r    fit)  dt 

As^o  Az  2TnJc{t  —  z^y* 

or 

^^"^^      2'iriJcit-zor' 
In  the  same  way,  we  may  show  that 

2!     C   fit)  at 

_  3!    r  fit)  at 
^  (^^  -  2^Jc  (^=15"*' 


f'^Kzo) 


n\     r     fix 
2  TTi  Jc  {t  — 


f{t)dt 


zoY 


The  existence  of  these  integrals  enables  us  to  affirm  the  existence  of 
the  higher  derivatives  of  /(z).  Consequently,  the  derivative  J'{z) 
is  continuous  and  holomorphic  as  the  theorem  states.  A  similar 
statement  may  now  be  made  with  reference  to  each  of  the  higher 
derivatives. 

Since/' (2)  is  holomorphic  in  S  iif(z)  is  holomorphic  in  S,  it  follows 
that  both  /(z)  and  f'{z)  are  continuous  in  any  closed  region  S'  lying 
within  S  and  hence  by  Theorem  III,  Art.  13,  both  f{z)  and  f'{z)  are 
uniformly  continuous  in  ;S'. 

The  fact  that  the  continuity  of  the  derivative /'(z)  follows  from  its 
existence  renders  the  theory  of  analytic  functions  of  a  complex  variable 
in  many  respects  simpler  than  the  theory  of  functions  of  a  real  vari- 
able; for,  a  derivative  of  a  function  of  a  real  variable  may  exist  at 
every  point  in  an  interval  and  yet  not  be  continuous  throughout 
the  interval.  In  the  next  article,  it  will  be  shown  that  the  conti- 
nuity of  the  partial  derivatives  of  the  first  order  of  u,  v,  where  /(z)  = 
u(x,  y)  +  iv{x,  y),  follow  from  the  continuity  of /'(z).  When  that 
result  has  been  established,  we  shall  be  able  to  apply  to  subsequent 
discussions  the  results  of  Green's  theorem. 


80  DIFFERENTIATION   AND   INTEGRATION         [Chap.  III. 


Theorem  III.  Let  f(t)  he  a  continuous  function  of  the  complex 
variable  t  along  an  ordinary  curve  C,  which  may  be  dosed  or  not.     The 

integral 

ffit)dt 

Jet  —  z 

defines  a  function  of  z  which  is  holomorphic  for  all  values  of  z  different 
from  t. 

It  is  at  once  evident  that  the  given  integral  defines  a  function  of  z. 
We  may  put 

Then,  by  the  reasoning  employed  in  the  demonstration  of  Theorem  II, 
we  obtain 

that  is,  F{z)  has  a  derivative  for  each  value  of  z  different  from  t. 
Hence,  the  function  F(z)  is  holomorphic  for  all  such  values  of  z  and 
if  F(z)  has  any  singular  points,  they  must  be  points  on  the  curve  C. 

By  aid  of  the  foregoing  theorems,  we  can  now  prove  the  converse 
of  the  Cauchy-Goursat  theorem.  This  theorem  is  due  to  Morera  * 
and  may  be  stated  as  follows: 

Theorem  IV.     If  f{z)  is  coniinuous  in  a  given  region  S  and  if  the 

integral    j  f{z)  dz  is  zero  when  taken  around  the  complete  boundary  C 

of  any  portion  of  S,  such  that  C  lies  wholly  within  S  and  incloses  only 
points  of  S,  then  f{z)  is  holomorphic  in  S. 

If  the  given  region  S  is  multiply  connected,  let  it  be  made  simply 
connected  by  the  introduction  of  cross-cuts.  Then  every  closed 
curve  C  lying  within  the  new  region  S'  is  a  complete  boundarj'^  and 
by  hypothesis  the  integral  taken  along  such  a  curve  is  zero.  We  shall 
show  that  in  this  simply  connected  region  f(z)  is  holomorphic  and 
hence  holomorphic  in  *S,  even  though  this  given  region  is  multiply 
connected.  Let  a  be  a  fixed  point  of  S  and  Zo  any  other  point  of  the 
same  region.  Denote  hy  Zo  -\-  Az  any  point  of  *S'  in  the  neighborhood 
of  Zo.    Because 

^f{z)  dz  =  0, 


X- 


'C 

•  See  Reale  InstUulo  Lombardo  di  sdeme  e  letlere,  Rendiconli,  2  series,  Vol.  19 
(1886),  p.  304. 


Abt.  20.] 


CAUCHY'S  INTEGRAL 


81 


it  follows  from  Theorem  III,  Art.  19,  that  the  value  of  the  integral 
between  any  two  points  is  independent  of  the  path.  Since  we  are 
at  liberty,  therefore,  to  select  arbitrarily  the  path  of  integration 
between  a  and  Zo  +  Az,  without  affecting  the  value  of  the  integral 

j         f(z)  dz,  we  take  a  path  pass- 

ing  through  Zq  and  rectilinear  be- 
tween 2o  and  Zq  +  Az,  as  indicated  in 
Fig.  32.     The  value  of  the  integral 

/   /(«)  dz,  where  f  is  a  point  upon  this 

path,  is  a  function  of  f ,  and  we  may 
write 

Hence,  we  have  for  f  =Zo  and  f  =Zo  +  Ae  the  following  values  of  F{^) : 

F{z,)=  £f{z)dz, 
F{zo  +  Az)  =  J^       fiz)  dz, 


whence 


'         f{z)dz-    /    f{z) 

a  t/a 


dz 


'Zo+Az 


Kz)  dz. 


(12) 


The  given  function  f(z)  is  continuous  in  S,  and,  therefore,  we  have 
for  an  arbitrarily  small  positive  number  e,  another  positive  number  5 
such  that 

1/(2)  -/(^o)  I  <  6,  for   \z-Zo\^\Az\<5. 

This  relation  may  be  written 

fiz)=f(z,)-\-n{z), 

where 

I  T)  I  <  €,  for    I  A  2;  I  <  5. 

Putting  f(zo)  +  T]  in  place  of  f(z)  we  obtain  from  (12) 

f{zo)dz+    I  Ti  dz. 


(13) 


82 


DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 


Dividing  by  Az,  we  have 
Az       Az 


f{zo)dz-\- 


1      /V,H 

which  is  equal  to 


/(2o)  dz  = 


Az 


ridz. 


(14) 


«0 

rt-A« 


dz, 


Az 


Az=f{zo). 


Moreover,  we  have 


-f 


'«*+A* 


-r^dz 


'Zc+Ai 


dz 


Zo+Ai 


From  (14)  we  now  get 

AF 
Az 


-/(^o) 


Az 


<e, 


dz 


Az  1  <  5; 


that  is, 


^      L^^nzo)=/(zo). 


Since  Zo  was  taken  to  be  any  point  of  S,  it  follows  that  for  all  values 
of  z  in  <S  we  have 

F'(z)=/(z).  (15) 

Consequently,  F{z)  is  holomorphic  in  S;  but,  as  we  have  seen,  the 
derivative  of  such  a  function  is  also  holomorphic;  hence  /(z)  must 
likewise  be  holomorphic  in  S,  as  the  theorem  requires. 

The  Cauchy-Goursat  theorem  states  a  necessary  condition  that  a 
given  function  /(z)  shall  be  holomorphic  in  a  given  region.  The 
theorem  just  demonstrated  gives  a  sufficient  condition  that  a  con- 
tinuous function  is  holomorphic.  We  may  combine  these  two  results 
into  the  following  theorem. 

Theorem  V.  The  necessary  and  sufficient  condition  that  a  continuous 
function  f(z)  is  holomorphic  in  a  given  finite  region  S  is  that  the  integral 

I  /(z)  dz  is  zero  when  taken  along  the  complete  boundary  C  of  any  portion 

of  the  plane  when  C  lies  entirely  within  S  and  incloses  only  points  of  S. 

21.  Cauchy-Riemann  differential  equations.  In  Art.  20  we  dis- 
cussed the  necessary  and  sufficient  condition  that  a  function  /(z) 


Art.  21.]  CAUCHY-RIEMANN  EQUATIONS  83 

is  holomorphic  in  a  given  JBinite  region  S.  This  condition  was  ex- 
pressed in  terms  of  a  definite  integral.  It  is  often  more  convenient 
to  have  such  a  condition  expressed  in  terms  of  the  partial  derivatives 
of  u  and  v,  where 

f(z)  =w  =  u{x,  y)  +  iv{x,  y) 

is  the  given  function.  Such  a  criterion  is  given  in  the  following 
theorem. 

Theorem  I.  In  a  given  finite  region  S  let  u  and  v  he  two  real  single- 
valued  functions  of  the  real  variables  x,  y.  The  necessary  and  sufficient 
condition  that  the  complex  function 

w  =  u  -{-  iv 

is  holomorphic  in  S  is  that  the  partial  derivatives  of  u  and  v  of  the  first 
order  exist  and  are  continvxms  and  moreover  satisfy  the  following  partial 
differential  eqacdions: 

du  _dv_  du  _      dv  ,  . 

dx~  dy'        dy~~dx'  ^*^ 

These  differential  equations  are  known  as  the  Cauchy-Riemann 
differential  equations.  To  show  that  these  equations  present  a  neces- 
sary condition  we  proceed  as  follows.  Since  the  function  w  is  holo- 
morphic in  aS,  it  has  a  derivative  with  respect  to  z.  As  we  have  seen, 
the  existence  of  this  derivative  involves  the  condition  that  the  ratio 

-—  shall  have  the  same  limiting  value  as  Az  approaches  zero  in  any 

direction  whatsoever.  Consequently,  the  same  limiting  value  is  ob- 
tained if  Az  is  permitted  to  approach  zero  through  real  values  or 
through  purely  imaginary  values.  The  increment  Az  =  Ax -\-  i  Ay 
becomes  in  the  first  case  Ax  and  in  the  second  case  i  Ay.  We  may 
therefore  write 

y.    Aw        J.    Aw       J    1  Aw  ,  . 

Li    — —  =     Li    -T —  =     Li     -  — —  •  IZ) 

As=o  Az      Ai=o  Ax      Ay=o  I  Ay 

Since  the  first  limit  exists  by  hypothesis,  the  second  and  third  limits 
must  also  exist.     We  have  then 

dw  _  dw  __  1  dw  ,„. 

dz       dx      i  dy 


84  DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 

However,  we  have 

^  =  1^  [u{x,  y)  +  iv{x,  2/)]  =  i  +  *|,  (4) 

dw       d  r   ,       X    ,    .  /       M       du   ,    .dv  ,-x 

^  =  ^Kx,y)  +  «;(x,i/)]  =  ^  +  t^-  (5) 

Substituting  these  values  in  (3),  we  obtain 

du  ,    .dv  .du   ,   dv  ,„. 

3-  +  t3-  =  - iT-  +  — •  (6) 

dx        dx  dy       dy  , 

Equating  the  real  parts  and  the  imaginary  parts  in  this  equation,  we 

have 

du  _dv  du  _  _dv^  ....s 

dx~dy'         dy~~dx'  ^'^ 

By  Theorem  II,  Art.  20,  the  derivative  -r-  is  continuous.  The  con- 
tinuity of  the  partial  derivatives  in  (7)  follows  therefore  from  equa- 
tions (3),  (4),  and  (5)  and  Theorem  II,  Art.  13. 

We  may  now  show  the  conditions  of  the  theorem  to  be  sufl&cient 
as  follows.     From  equation  (2),  Art.  17,  we  have 

I  /(«)  dz  ==  j  u  dx  —  V  dy  -\-  i  j  V  dx  -\-  u  dy,  (8) 

where  C  may  be  regarded  as  any  path  of  integration  within  the  given 
region  S.    As  u,  v  are  continuous,  both  of  the  integrals 

j  udx  —  vdy,  I  vdx  +  udy  (9) 

exist.  By  hypothesis,  the  equations  (1)  are  satisfied  by  u,  v.  Hence, 
by  Theorem  II,  Art.  16,  both  of  the  integrals  in  (9)  are  zero,  and  we 
have  from  (8) 

^  f{z)dz  =  0 


L 


for  every  path  of  integration  C  forming  a  complete  boundary  of  any 
portion  of  S  such  that  C  hes  entirely  within  S  and  incloses  only 
points  of  S.    Consequently,  by  Theorem  IV,  Art.  20,  /(z)  is  holo- 
morphic  in  the  given  region  ;S  as  the  theorem  requires. 
From  equations  (3),  (4),  and  (5),  we  have 

dw^  _du       .dv  _dv_       .du 
dz~di'^''di~di~''di'  ^^"^ 


Akt.  21.1  CAUCHY-RIEMANN  EQUATIONS  85 

This  relation  affords  a  convenient  method  of  computing  the  deriv- 
ative of  w,  when  w  =  f(z)  is  expressed  in  tenns  of  x  and  y. 

Ex.   Given 

w  =  x^  -\-  3  x-yi  —  3  xy'^  —  yH. 

Show  that  w  is  holomorphic  everywhere  in  the  finite  region,  and  compute  DzW. 
We  have 

u  =  x'  —  3  xy"^,  V  ==  Zxhf  —  y', 

?^=-6xy,  |?^  =  3x^-32/^ 

dy  "'  dy 

The  conditions  of  the  theorem  are  fulfilled  and  the  function  is  therefore  holo- 
morphic for  all  finite  values  of  x  and  y.    For  the  derivative  DzW  we  have 

DzW  =  ^  +  i  ^  =  3  x2  -  3 1/2  -I-  6  ixy. 

In  this  particular  case  w  can  be  readily  expressed  directly  as  a  function  of  z, 
namely  w  =  z^.  We  have  then  by  the  laws  of  differentiation  DzW  =  3  2-,  a 
result  which  is  identical  with  that  already  obtained. 

In  the  calculus  of  real  variables  and  in  trigonometry,  we  became 
acquainted  with  certain  pairs  of  functions  that  we  called  inverse 
functions.  For  example,  i/  =  sin  x  and  x  =  arc  &vciy,  y  =  x^  and 
X  =  Vy  are  illustrations  of  inverse  functions.  If  we  have  given  any 
function  w  =  /(z),  z  may  be  regarded  as  a  function  of  w,  expressed 
implicitly  by  this  equation  in  w  and  z.  It  is  desirable,  however,  to 
know  the  conditions  under  which  z,  considered  as  a  function  of  w,  is 
holomorphic  in  a  definite  region  when  /(z)  is  holomorphic  in  a  given 
region.     The  desired  conditions  may  be  stated  as  follows : 

Theorem  II.  Lei  w  =  f{z)  he  holomorphic  in  a  given  region  T, 
which  is  defined  by  the  inequality  \  z  —  Zo  \  <  h.  Moreover,  let  f'{z)  7^  0 
joT  values  of  z  in  T  and  let  Wq  =  S{zo).  Then  corresponding  to  the 
number  h  there  is  determined  a  number  k  such  that  in  the  W-plane  there 
exists  a  region  S,  defined  by  the  inequality  \w  —  Wq  \  <  k,  for  values 
of  w  within  which  the  equation 

w  =  f(z), 
has  one  and  only  one  solution 

z  =  F(w); 

mxrreover  the  inverse  function  thus  determined  is  holomorphic  in  S  and 


86  DIFFERENTIATION  AND  INTEGRATION         [Chap.  Ill, 

We  have 

w  =  f(z)  =  u  +  iv. 

The  functions  w,  v  have  continuous  first  derivatives  which  satisfy 
simultaneously  the  equations: 


du 
dx 

Consider  now  the  Jacobian 


dv 
dy 


du 
By 


dx 


(11) 


(12) 


By  aid  of  the  conditional  equations  (11),  this  determinant  can  be 
written  in  the  form 


du 
dx 

du 

dy 

dv 

dv 

dx 

dy 

du 

dx 

dv 

dx 

dv 

du 

dx 

dx 

-(MfHW- 


du        .dv 
dx  dx 


=  I  f'iz)  \'- 


By  hypothesis  f'{z)  is  different  from  zero,  and,  hence,  the  Jacobian 
does  not  vanish.  The  non-vanishing  of  this  determinant  is,  how- 
ever, the  condition  that  a  region  S  of  the  TF-plane  exists,  for  every 
point  of  which  the  two  equations 

u  =  ^i(x,  y),     v  =  ^2(a:,  y) 

have  one  and  only  one  simultaneous  solution,  expressing  x  and  y  in 
terms  of  u,  v,  say 

x  =  xi(u,v),    y  =  X2(u,v), 

where  these  new  functions  of  u,  v  have  continuous  first  derivatives 
with  respect  to  u  and  to  v.*    We  may  then  write 

z  =  x-^iy  =  xi(w,  v)  +  ixi(u,  v)  =  F{w). 

It  still  remains,  however,  to  show  that  the  function 

z  =  F{w), 

found  in  this  way,  is  holomorphic  in  the  region  S. 

*  See  Erwyklopddie  der  Math.  Wissenschaften,  Vol.  II,  Heft  1,  Art.  5;    also 
GouTsat-Hedrick,  Mathematical  Analysis,  Vol.  I,  p.  50. 


Art.  21.]  CAUCHY-RIEMANN  EQUATIONS  87 

We  have  for  Aw,  Az,  each  different  from  zero, 

^  =  ^-  (13) 

Aw      Aw  ^    ' 

We  wish  to  consider  the  hmit  as  Aw;  approaches  zero.  Since  F{w) 
is  continuous,  we  know  that  Az  approaches  zero  simultaneously 
with  Aw.    We  may  therefore  write 

i  ^=  ix-  (1*) 

Ato=o  Aw      Az=o  Aw 
~Az 

But  as  j\£)  =  L  ——  is  different  from  zero,  we  have 
A2=o  A2; 

L  ^  =   L  -i- = -I--  (15) 

Au,=oAw;      Az=oAu?         J    Aw 

Az       Az=o  Az 
In  this  equation  we  know  that  the  limit  in  the  right-hand  member 
exists  and  defines  ipj-r ,  because  w  =  f{z)  is  by  hypothesis  holo- 

morphic  in  the  given  region  T.    It  foUows  from  (13)  that  the  limit 

Az 
L  ——  must  also  exist  for  all  points  in  S;  that  is,  z  —  F(w)  has  a 

Au,=s=0  Aw 

derivative  for  all  values  oi  w  in.  S.  Hence,  F{w)  is  holomorphic  in  S 
as  the  theorem  requires. 

From  (15)  we  have  the  relation 

F'(w)=j^y  (16) 

between  the  derivative  of  the  given  function  and  that  of  its  inverse, 
which  was  also  to  be  demonstrated. 

In  Art.  20,  it  was  shown  that  when  f(z)  is  holomorphic,  f(z)  is 
necessarily  continuous.  It  can  now  be  demonstrated  that  the  differ- 
ence quotient  converges  uniformly  to  the  derivative  as  stated  in  the 
following  theorem. 

Theorem  III.  If  f{z)  is  holomorphic  in  a  finite  closed  region  S, 
then  the  difference  quotient 

f(z-\-Az)-f{z) 
Az 

converges  uniformly  to  the  limit  f'{z)  for  values  of  z  in  S. 

This  theorem  is  equivalent  to  saying  that  if  /(«)  is  holomorphic  in 


88  DIFFERENTIATION  AND  INTEGRATION         (Chap.  III. 

S  then  for  an  arbitrarily  chosen  positive  number  c,  there  exists 
another  positive  number  r,  such  that 

fiz-hAz)-f(z) 


-S'iz) 


<  €,  I  A2  I   <  T)  (17) 


^z 

for  all  values  of  2  in  <S.  In  other  words,  the  value  of  e  being  deter- 
mined arbitrarily  there  exists  a  number  tq  independent  of  z  satisfying 
the  required  condition.     We  may  put 

z  =  x  +  iy,       4^=  u(x,  y)  +  iv{x,  y), 
and  hence  have  T^  v 

f{z  +  Az)  -  f(z)  =  Afiz)  =  Au  +  i  Av,         Az  =  Ax  +  i  Ay.      (18) 
Since  w  is  a  function  of  x  and  y,  we  may  write 
Am  =  u(x  -\-  Ax,    y  +  Ay)  —  u(x,  y) 

=  u{x-\-  Ax,     y-\-  Ay)  -  u{x-\-Ax,  ?/)+w(x+  Ax,  y)  -  u{x,  y). 

By  making  use  of  the  law  of  the  mean,  this  result  may  be  written 
Am  =  Uy\x  +  Ax,  y  -\-d\  Ay)  Ay  +  uJix  +  ^2  Ax,  y)  Ax, 

0  <  01  <  1,        0  <  02  <  1,  (19) 

where  uj,  uj  denote  partial  derivatives  with  respect  to  x  and  y, 
respectively.     In  a  similar  manner,  we  may  get 

Av  =  Vy'ix  +  Ax,y  -{-Bz  Ay)  Ay  +  vj{x  +  6^  Ax,  y)  Ax, 

0  <  03  <  1,        0  <  04  <  1.  (20) 

We  have  seen,  under  Theorem  I,  that 

S\z)  =  uj{x,  y)  +  iv;r\x,  y). 

By  aid  of  (19)  and  (20)  and  the  Cauchy-Riemann  differential  equa- 
tions, we  may  now  write 
f(z-^Az)-f(z)  _  ^,^^^  ^  f(z  +Az)-  f(z)  -  Azf'iz) 


Az 

Au  -j-  i  Av  —  {Ax  -\-  i  Ay)  {ux'-\-  ivj) 
— 

Ay 


=  [uy{x -{-Ax,y-\-  01  Ay)—Uy{x,y)] 


Ax-\-iAy   ■ 

Ax 
+  [uJix  -\-  02  Ax,  y)  -  uj{x,  y)]^_^^. 

+t[v/(x+Ax,  y-\-hAy)-Vy'{x,  y)\  ^^^ 

+  iW{x  -H  04  Ax,  y)  -  vJix,  y)\  -^^.    • 


Art.  22.]  CHANGE  OF  VARIABLE  89 

Since  uj ,  Uy,  vj,  Vy  are  continuous  in  *S  they  are  uniformly  con- 
tinuous in  this  region  *  and  each  of  the  expressions  inclosed  in 

brackets  can  consequently  be  made  less  in  absolute  value  than  j  by 

the  proper  choice  of  Ax,  Ay;  for  example,  if  Ax,  Ay  be  so  taken  that 
Vax^  +  Ay2  <  Tj.  The  absolute  value  of  each  of  the  factors  out- 
side of  the  brackets  is  never  greater  than  unity.  Consequently,  we 
may  write 

f{z  +  ^z)  -  f(z) 


Az 


-f'iz) 


<  e,         VAx2  +  Ay2  =  I  A2  I  <  Tj. 


Hence  the  theorem. 

22.  Change  of  complex  variable.  In  Art.  18,  it  was  pointed  out 
that  the  law  for  the  change  of  variable  in  a  definite  integral  of  a 
function  of  a  real  variable  can  be  extended  without  modification 
to  the  case  of  a  definite  integral  of  a  function  of  a  complex  variable 
where  the  change  is  from  a  complex  variable  to  a  real  variable.  We 
can  now  complete  the  discussion  by  showing  that  the  same  law  holds 
where  the  change  is  from  one  complex  variable  to  another. 

Suppose  we  put 

z  =  <t>(t), 

where  t  is  complex  and  <l>{t)  is  holomorphic  along  a  curve  K.  As  t 
traces  the  curve  K,  suppose  the  variable  z  traces  the  path  C.  Cor- 
responding to  the  points  of  division  U,  h,  .  .  .  ,  tn  upon  K,  we  have 
the  points  Zo,  Zi,  .  .  .  ,  0„  of  division  of  C.  Since  <f}{t)  is  holomorphic 
along  K  and  hence  has  a  derivative  <t>'{t)  along  K,  we  have 

^  =  |^£  =  <^'(«^-0  +  ^*>        A:  =  0,  1,  2,  .  .  .  n,        (1) 

where  e*  vanishes  with  A^^.     If  in  the  sum 

we  replace  A^z  by  its  value  obtained  from  (1),  we  have 

Since  z  =  <i>(t)  is  by  hypothesis  holomorphic  along  K,  it  follows  from 
Theorem  III,  Art.  21,  that  corresponding  to  an  arbitrarily  small 

*  See  Goursat-Hedrick,  Mathematical  Analysis,  Vol.  I,  p.  251. 


90  DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 

positive  nutober  e  there  exists  a  positive  number  5,  such  that  for 
I  Afc/  I  <  6  we  have 

that  is,  the  various  values  of  e*  can  be  replaced  by  a  single  arbitrarily 
small  value  c,  if  the  values  of  ^i^  are  all  taken  less  in  absolute  value 
than  8.    We  then  have 

I  X^kf{zk-i)  Akt I  =  eM^)  \Akt\  =  tML,  (3) 

where  M  is  the  maximum  value  of  |  /(z)  |  upon  C  and  L  is  the  length 
of  K.  Smce  ML  is  a  constant  and  e  is  arbitrarily  small  the  limit  of 
the  sum  in  (3)  is  zero  as  At  approaches  zero;  hence,  we  have  from 
(2)  and  (3)  upon  passing  to  the  limit 

ff(z)dz=    f  f\m\^'(*)dt,  (4) 

Jc  Jk 

which  expresses  the  law  for  the  change  of  variable  for  the  case  under 
consideration. 

Ex.  Given  a  region  S  consisting  of  that  portion  of  the  complex  plane  exterior 
to  its  boundary  C,  which  is  taken  to  be  a  closed  curve  exterior  to  the  unit  circle 

about  the  origin.    Consider  the  integral  of  /(«)  =  -  taken  over  the  boundary  C 

of  the  region  S. 

By  putting  z  =  -,  C  goes  into  a  curve  K  about  the  origin  and  Ijring  within 
z 

the  unit  circle.    We  have 

=  -  C  z'dz'  =  0.  (Th.  5Sr,  Art.  20.) 

It  will  be  observed  that  while  the  first  integral  must  be  taken  in  a  clockwise 
direction,  as  the  region  S  with  respect  to  which  the  integral  is  taken  lies  exterior 
to  the  curve  C,  the  integral  along  the  curve  K  is  taken  in  a  counter-clockwise 

direction;  for,  as  z  traces  out  C  in  a  clockwise  direction  -  traces  out  X  in  a  counter- 
clockwise  direction. 

23.  Indefinite  integrals.  Let  f{z)  be  holomorphic  in  a  given 
finite  region  S.  As  we  have  seen,  the  integral  i  f(z)  dz  defines  in 
S  a  function  F{z)  of  the  variable  limit  of  integration.    From  the  dis- 


Art.  23.]  INDEFINITE  INTEGRALS  91 

cussion  of  Theorem  HH^  of  Art.  20,  it  follows  that  this  function 
F{z)  is  also  holomorphic  in  *S,  and  furthermore  that  the  relation 
between  f{z)  and  F{z)  is  such  that 

dF{z) 


/(2)  = 


dz 


Let  <i>{z)  denote  any  function  having  the  derivative  f{z).  Such  a 
function  is  called  a  primitive  f tmction  of  f{z) .  We  write  as  in  the  cal- 
culus of  real  variables 

<l>{z)  =  j  f{z)  dz, 

and  speak  of  /  /(z)  dz  as  the  indefinite  integral. 

The  primitive  function  (t>(z)  can  differ  from  F(z)  at  most  by  a 
constant.  For,  (l>{z)  is  holomorphic  in  S,  since  it  has  a  derivative 
f{z),  and  hence  F(z)  —  <t>{z)  is  also  holomorphic  in  S.  We  have, 
since  both  0(z)  and  F{z)  are  primitive  functions  of /(z), 

|[n.)-*Wl=f-g=/W-/W=o  (5) 

for  all  values  of  z  in  <S.    We  may  write 

F{z)  -  <^(z)  =  u{x,  y)  -\-  iv{x,  y), 
and  have  from  Art.  21 

From  (5)  and  (6)  it  follows  that 


du  ,    .dv       _ 


and  hence  we  must  have 


From  the  Cauchy-Riemann  differential  equations,  we  have  also 

5^  =  0.        r  =  0.  (8) 

dy        '         dy 

As  (7)  and  (8)  hold  for  all  values  of  (x,  y)  in  S,  it  follows  that  both 
u  and  V  are  real  constants.*     Consequently,  u  +  iv  must  be  a  complex 

*  Compare  Pierpont,  Theory  of  Functions  of  Real  Variables,  Vol.  I,  p.  250. 


"92  DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 

constant  and  F{z)  differs  from  4>{z)  by  this  constant  value  for  all 
values  of  z  in  »S. 

Since  <l>{z)  differs  from  F{z)  by  a  constant  for  all  values  of  z  along 
the  path  of  integration  between  any  two  points  a,  fi  of  S,  we  may 
write 

<t>{z)  =  Jf(z)  dz  +  c. 

For  z  =  a,  this  relation  becomes 

<t>(a)  =  C. 

If,  on  the  other  hand,  z  =  /3,  we  get 

<^(/3)  =Jj(z)dz  +  c. 

From  these  two  results,  we  have  at  once  the  fundamental  theorem  of 
the  integral  calculus,  namely: 

jy{z)dz  =  ^{^)-<^(a);  (9) 

that  is,  the  law  for  the  evaluation  of  a  definite  integral  in  the  calculus 
of  real  variables  may  be  extended  without  modification  to  functions 
which  are  holomorphic  in  a  given  finite  region. 

Ex.   Given  the  function  /(«)  =  2"*,  m  5^—  1;   find  the  value  of  the  integral 

/(z)  dz  =  -^-—r  +  c. 
m  -\-  y. 

Hence,  from  (1)  we  obtain 

gm+i   -12,        g^m+i  _  g^w+i 


w  +  1 


24.   Laplace's    differential    equation.    We    have   the   following 
theorem. 

Theorem  I.     In  a  given  finite  region  S,  let  the  complex  function 

f(z)  =  u-\-iv 

be  holomorphic;  then  the  functions  u(x,  y),  v{x,  y)  satisfy  the  partial 
differential  equaiion 

—  +  —  =  0  m 


Art.  24.]  LAPLACE'S  EQUATION  93 

This  differential  equation  is  known  as  Laplace's  differential  equa- 
tion and  is  of  prime  importance  in  theoretical  physics. 

Since /(z)  is  holomorphic  in  S,  the  derivative /'(z)  and  also  the  higher 
derivatives  exist  and  are  holomorphic  in  the  same  region.  It  follows 
that  the  partial  derivatives  of  u,  v  with  respect  to  x  and  y  exist  and 
are  continuous.  This  statement  holds  not  only  for  the  partial 
derivatives  of  the  first  order  but  likewise  for  those  of  the  second  and 
higher  orders.     From  the  Cauchy-Riemann  differential  equations, 

(2) 


du       dv 
dx~  dy' 

du 
d^~ 

_di 
dx' 

we  obtain  by  differentiation 

d^u        d^v 

dhi 

dH 

dx^      dxdy' 

dy' 

dy  dx 

(3) 

As  the  partial  derivatives  of  the  second  order  are  continuous  in 
X,  y,  together,  we  have  * 

dh)  dH 


dx  dy       dy  dx 


(4) 


Hence,  by  addition  of  the  equations  Ci^,  we  have 

S  +  I^^o-  (^) 

In  a  similar  manner  we  can  show  that 

dH       5^  _  „ 
dx^      dy' 

We  shall  now  consider  the  converse  proposition,  namely: 

Theorem  II.  //  in  a  given  finite  region  S,  a  function  u{x,  y)  has 
continuous  partial  derivatives  of  the  first  and  second  order  and  satisfies 
Laplace's  differential  equation,  then  there  exists  a  function  v(x,  y)  de- 
termined except  as  to  an  additive  constant,  such  that  the  complex  function 
u  '-\-  iv  =  f{z)  is  holomorphic  in  S. 

We  have  given  the  condition  that  u  satisfies  the  differential  equa- 
tion 

d'u      dhi  _  - 
d^'^dy^~ 

*  See  Townsend  and  Goodenough,  First  Course  in  Calculus,  Art.  104. 


94  DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 

We  now  define  v  by  the  relation 

This  integral  exists  because  the  integrand  is  continuous,  since  — ' 

-r—  are  continuous.     Moreover,  the  integral  is  independent  of  the 
path  by  virtue  of  Theorem  III,  Art.  16,  because 

dy[     dy)~dx\dx)' 


that  is, 


dht      dhi  _  „ 
dy^^'dx^~ 


Differentiating  v  partially  with  respect  to  x,  we  have  from  (6)  by  aid 
of  Theorem  IV,  Art.  16, 

dx  dy  '  ^^ 

Similarly,  differentiating  with  respect  to  y,  we  get 

dy      dx'  ^  ^ 

Equations  (7),  (8)  are  however  none  other  than  the  Cauchy-Riemann 
differential  equations,  and  hence 

u  +  iv  =  f(z) 
is  holomorphic  in  S. 

Ex.  1.  Given  u  =  x^  —  3  xy^.  Show  that  there  exists  a  function  v{x,  y)  such 
that  w  =  u  -^  iv  is  holomorphic  in  the  finite  region.  Determine  the  function 
v{x,  y). 

The  given  fvmction  satisfies  Laplace's  equation;  for,  we  have 

=  -6xt/, 
=  -6a;, 

oiir  oy- 

hence 


The  required  value  of  »  is  to  be  determined  from  equation  (6).  As  pointed  out 
in  the  discussion  of  the  foregoing  theorem,  this  integral  is  independent  of  the 
path.    It  can  be  conveniently  evaluated  by  making  the  path  rectilinear  passing 


du 
dx 

3x2 

-3 

y\ 

du 

dy 

dx* 

6z, 

dhl 
dy^ 

dx-' 

^  dy^ 

=  0. 

Art.  24.]  LAPLACE'S  EQUATION  95 

from  (xo,  2/0)  to  (x,  yo),  thence  to  (x,  y).  From  (xo,  2/0)  to  (x,  yo)  we  have  y  =  j/o, 
dy  =  0,  while  X  varies  from  xo  to  x.  From  (x,  7/0)  to  (x,  y),  we  have  dx  =  0  and  y 
varying  from  yo  to  y.     Hence,  from  equation  (6)  we  have 

V  =    f  6xyodx+  C   (3  x2  -  3  y2)  dy  +  C 

=  'S^a-^a  -  3  XoVo  +  3  x^y  -  y3  -  3  x^yo  +yo^  +  C 
=  3  x2y  -  y3  -  (3  Xo^'yo  -  yo')  +  C. 

Putting  C  —  3  Xo'^yo  +  yo'  =  c, 

we  have  v  =  S  x^y  —  y^  +  c, 


whence 


f{z)  =  u  +  iv  =  x^  —  3  xy'^  +  i  \  {3  x^y  —  y^)  +  c]. 
=  (x  +  iyy  +  ic  =  z^  +  ic. 


From  the  form  of /(z),  it  will  be  at  once  seen  that  it  is  holomorphic  in  any  finite 
region. 

Ex.  2.     Given  u  =  log  (x^  +  y^)  .     Find  a  function  v{x,  y)  such  that  u  +  iv 
is  an  analytic  fimction. 
We  have 

du  _        X  du  _        y 

dx  ~  WT¥) '  dy  ~  (x2  +  y*) ' 

dht  _    y2  -  x2  dhi        x2  -  y2 


ax2       (x2  +  y2)2  dy'^       (x2  +  y2)2 

These  results  substituted  in  Laplace's  equation  show  that  u  satisfies  that  equa- 
tion. As  in  Ex.  1,  it  is  convenient  to  take  the  path  as  rectilinear  through  the 
intermediate  point  (x,  yo).     From  equation  (6),  we  then  have 


"=    r-T:L^dx+   r^^dy  +  C 
Jxc  x2  +  yo*  Jv„  x"  +  y2 


+  yo*  Jy^  x*  +  y2 

=  —  arc  tan \-  arc  tan  —  +  arc  tan  -  —  arc  tan  —  +  C 

yo  yo  a;  X 

=  arc  tan  -  —  ^  +  arc  tan  —  +  C. 
X      2  '  yo 


If  we  now  put 


C  —  7:  +  arc  tan  —  =  c, 

2  yo 


we  have  v  =  arc  tan  -  +  c, 

X 

and  hence  get 

1  y 

w  +  it;  =  log  (x^  +  y^)    +  i  arc  tan  -  +  ic. 

X 

The  function  w  =  f{z)  =  u  +  iv 

is  holomorphic  for  aU  values  of  z  in  any  finite  region  not  including  the  origin, 
since  for  such  values  oi  z  =  x  +  iy  the  functions  u  and  v  satisfy  the  conditions 
of  Theorem  II. 


96  DIFFERENTIATION  AND  INTEGRATION         [Chap.  III. 

25.  Applications  to  physics.  A  variety  of  problems  in  mathe- 
matical physics  are  associated  with  the  solution  of  Laplace's  equation. 
According  to  Newton's  law  two  bodies  in  space  attract  each  other 
directly  as  the  product  of  their  masses  and  inversely  as  the  square 
of  the  distance  between  them.  When  one  of  these  bodies  is  moved 
with  respect  to  the  other,  then  work  is  done  in  overcoming  the 
attractive  force  of  the  second  body.  The  work  done  in  overcom- 
ing the  attractive  force  of  a  given  mass  M  so  as  to  move  a  particle 
of  unit  mass  from  a  given  point  to  an  infinite  distance  is  defined  as 
the  potential  of  M  at  that  point.  It  can  be  shown  that  the  poten- 
tial is  a  function  of  the  space  coordinates  x,  y,  z  alone;  that  is,  that 
it  is  independent  of  the  path.  A  Newtonian  potential  function 
u(x,  y,  z)  due  to  attractive  matter  is  such  that  Laplace's  equation, 

dhi      dhi      dhi  _ 

must  be  satisfied  whenever  {x,  y,  z)  are  the  coordinates  of  a  point 
exterior  to  the  matter  itself. 

If  the  conditions  are  such  that  the  attractive  force  acts  only  in  a, 
plane,  taken  conveniently  as  the  XF-plane,  then  the  third  com- 

ponent  of  the  force  becomes  zero  and  -^  vanishes.    Consequently, 

for  two  dimensions  Laplace's  equation  takes  the  form  discussed  in 
the  preceding  article,  namely 

dhi      dhi  _ 

3x2  "*"  dy^  ~ 

In  this  case  the  potential  is  a  logarithmic  potential,  and  the  force 
overcome  varies  directly  as  the  product  of  the  masses  of  the  particles 
and  inversely  as  the  distance  between  them. 

From  the  discussion  in  the  last  article,  it  follows  that  if  m  is  a 
logarithmic  potential,  there  exists  a  function  v,  determined  except  as 
to  an  additive  constant,  such  that 

w  =  u{x,  y)  +  iv{x,  y), 

considered  as  a  function  of  z,  is  holomorphic  in  the  region  for  which 
the  potential  is  determined. 

A  potential  also  exists  in  connection  with  a  magnetic  or  electric 
field.     An  electric  potential  at  apy  point  may  be  defined  as  the  work 


Art.  25.1  APPLICATIONS  97 

necessary  to  be  done  against  an  electric  force  in  moving  a  unit  charge 
of  negative  electricity  from  that  point  to  an  infinite  distance.  The 
potential  may  be  defined  in  a  similar  manner  for  the  points  of  a 
magnetic  field.  In  any  case  the  derivatives  of  the  potential  with 
respect  to  x,  y  represent  the  components  of  the  force  in  the  direction 
of  the  two  axes.  In  order  to  have  the  two-dimensional  case  that 
arises  in  connection  with  the  discussion  of  functions  of  a  complex 
variable,  the  force  exerted  must  be  confined  to  a  plane,  taken  as  the 
complex  plane.  For  example,  such  a  case  arises  when  a  current  of 
electricity  flows  through  a  straight  wire  of  indefinite  length.  A  mag- 
netic field  is  created  in  the  surrounding  space  such  that  the  compo- 
nent of  the  force  in  the  direction  of  the  wire  is  zero.  Consequently 
any  plane  perpendicular  to  the  wire  may  be  taken  as  the  complex 
plane  and  the  application  reduces  to  one  in  two  dimensions. 

As  another  illustration  of  the  applications  of  the  functions  of  a 
complex  variable  may  be  mentioned  the  stationary  streaming  of  elec- 
tricity. Suppose,  for  example,  we  have  given  as  a  conductor  a  thin 
sheet  of  metal  of  unlimited  extent  and  of  uniform  thickness  and 
structure.  Let  the  current  of  electricity  be  introduced  into  and 
leave  this  conductor  by  means  of  perfectly  conducting  electrodes. 
The  current  may  then  be  regarded  as  flowing  in  a  plane  parallel  to 
the  two  surfaces  of  the  sheet.  The  illustration  then  becomes  a 
two-dimensional  one  and  the  condition  that  the  streaming  is 
stationary  is  that  for  each  value  of  (x,  y)  in  the  region  of  flow 
the  potential  is  such  that  Laplace's  equation  for  two  dimensions  is 
satisfied.* 

A  corresponding  application  to  the  flow  of  heat  may  be  readily  for- 
mulated. Let  the  body  in  which  the  flow  takes  place  be  a  cylinder 
of  indefinite  length  whose  rectangular  cross-section  consists  of  one 
or  more  closed  curves.  Upon  the  surface  of  this  cylinder  let  the 
temperature  w  be  a  constant  for  all  points  along  the  same  generator 
of  the  cylinder.  Moreover,  let  the  temperature  along  any  line 
parallel  to  a  generator  and  lying  within  the  cylinder  be  constant. 
Otherwise  let  the  temperature  vary  continuously  both  upon  the 
surface  and  within  the  cylinder.  The  flow  of  heat  then  takes  place 
in  planes  perpendicular  to  the  generators  of  the  cylinder.  The 
temperature  u  must  satisfy  Laplace's  equation  for  two  dimensions, 
if  the  flow  is  continuous. 

The  last  two  illustrations  are  special  cases  of  the  flow  of  incompres- 
*  See  Jeans,  Electricity  and  Magnetism,  Art.  389. 


98  DIFFERENTIATION   AND   INTEGRATION  [Chap.  III. 

sible  fluids.  If  w  is  a  function  of  the  space  coordinates  x,  y,  z  such 
that  the  components  of  the  velocity  of  the  fluid  are 

du  du  _  du 

~dx'  ~  dy'        ~Tz' 

then  u  is  called  a  velocity-potential  *  in  analogy  to  the  Newtonian 
potential  function  already  discussed.  The  existence  of  a  velocity- 
potential  is  a  property  not  of  a  region  of  space  but  of  portions  of 
matter.  As  the  portion  of  matter  moves  about,  it  carries  this  prop- 
erty with  it,  while  the  space  occupied  by  the  matter  at  any  instant 
may  come  to  be  occupied  by  matter  not  possessing  the  property. 
An  irrotational  motion  of  a  fluid  within  a  simply  connected  region  is 
characterized  by  the  existence  of  a  velocity-potential.  The  condi- 
tion that  the  given  fluid  flows  continuously  and  has  a  velocity- 
potential  u  is  that  u  satisfies  Laplace's  differential  equation.  If  we 
now  impose  such  conditions  upon  the  fluid  that  the  flow  takes  place 

in  a  plane,  then  r—  vanishes  and  the  problem  reduces  to  a  two- 
oz 

dimensional  one,  and  the  theory  of  functions  of  a  complex  variable 
may  be  applied.  To  accomplish  this  purpose,  let  us  suppose  that 
the  fluid  is  of  constant  density  and  flows  between  two  fixed  parallel 
planes  so  that  the  path  of  the  individual  points  of  the  fluid  lies  in  a 
plane  parallel  to  the  fixed  planes  and  the  fluid  flows  so  that  two 
points  which  at  any  instant  lie  in  a  line  perpendicular  to  the  fixed 
planes  remain  in  the  same  relative  position.  Then  any  plane  par- 
allel to  the  fixed  planes  may  be  taken  as  the  complex  plane  and  the 
theory  of  functions  of  a  complex  variable  becomes  at  once  applicable. 
In  the  next  chapter  we  shall  discuss  more  in  detail  some  examples 
illustrating  the  way  functions  of  a  complex  variable  may  be  employed 
in  particular  physical  problems. 

^    _         ,  EXERCISES 

I.  Show  that 

w  =  X*  -\-  4  ixhj  —  6  x^  —  4  ixy'  +  y* 

satisfies  the  conditions  given  in  Art.  21  for  finite  values  of  x  and  y  and  is  therefore 
holomorphic  in  the  finite  region.  Express  w  in  terms  of  z  and  compute  DzW  by 
the  method  given  in  that  article.  " —  'y-^-^" 

^  2.  Given /(z)  =  2",  where  n  is  a  positive  integer.  Find/(2o  +  Az)  —  f(zo)  by 
^mieans  of  the  binomial  theorem  and  show  for  all  values  of  z,  (a)  that  f{z)  is  con- 
tinuous, (6)  that /'(e)  =  rw"-*,  making  use  of  the  definition  of  a  derivative. 

*  For  fuller  discussion  of  the   properties  of  velocity-potentials,   see  Lamb, 
Hydrodynamics,  3^  Ed.,  Chapters  II  and  III. 


Art.  25.]  EXERCISES  99 

3.  By  making  use  of  the  definition  of  a  derivative  and  the  methods  employed 
in  the  calculus  of  real  variables  prove  that,  if  /,  /i,  ji  are  holomorphic  in  a  region 
R  and  hence  each  has  a  derivative  in  this  region,  the  following  laws  hold  for  values 
of  z  in  72: 

(a)  If  /  is  a  constant,  then  /'  =  0. 
(6)  If/  =/i  ±/2,  then/'  =//  ±/2'. 
(c)  If  /  =  /i  'h,  then  /'  =  U  'U  +h  •//• 

r 

^d)  If  /  =   j^,  where /2  t^  0  for  all  values  of  the  argument  considered,  then 

/  ^^    ,  , 

f,  _h  'h'  ~/i  'h' . 

4.  Show  that  every  rational  integral  function  of  z  is  holomorphic  in  the  entire 
finite  portion  of  the  complex  plane,  also  that  every  rational  function  of  z  is  holo- 
morphic in  any  region  not  including  points  where  the  denominator  is  zero. 

6.   Find  the  value  of  the  line-integral 

f  i3x  +  7y^)dx  +  {x^  +  Sy)dy, 

where  C  is  the  perimeter  of  a  square  whose  sides  are  x  =  0,  x  =  4,  y  =  2,  y  =  —  2. 
Is  this  line-integral  independent  of  the  path? 
6.   Evaluate  the  line-integral 


X 


(x2  +  7xy)dx  +  {3x  +  y^)  dy, 
C 

where  C  is  the  boimdary  of  the  multiply  connected  region  bounded  by  the  two 
curves  whose  equations  are  x^  -|-  y^  =  9,  (x  -f-  1)^  +  2/^  =  1. 
7.   Let  the  path  C  of  integration  be  given  by  the  equations 

X  =  3  cos  6,        y  =  2  sin  0. 
Find  the  value  of  the  integral 

'0,2 

(3x2  +  2xy  +  y^)dx, 


f 

•'3.( 


taken  along  the  path  C.    What  is  the  value  of  the  integral  of  the  given  fimction 
when  the  path  is  a  circle  about  the  origin? 

— *  8.   Evaluate  the  integral  f  (3  z"  +  7  z  -|-  9)  dz,  where  C  is  the  circle  x'+y^  =  3. 
Is  the  integrand  holomorphic  in  the  region  bounded  by  C? 

9.   Evaluate  the  integral  f  (2  z^  +  8  z  -|-  2)  dz,  where  C  is  the  arc  of  a  cycloid 

Jc 

y  =  a  {1  —  COS  e),  X  =  a  (e  —  sin  6)  between  (0,  0)  and  (2  7ra,  0). 

10.  Show  that  the  integrals  of  the  functions  given  in  Exs.  8,  9,  taken  about 
any  closed  curve  lying  in  the  finite  region  must  be  zero. 

11.  Is  the  integral 

2-3'-3z  +  7 


I 


dz 

3+2  J  Z 


independent  of  the  path  of  integration?     What  conclusion  can  be  drawn  from 
the  answer  as  to  the  nature  of  the  integrand  ? 


100  DIFFERENTIATION  AND  INTEGRATION        [Chap.  III. 

gi  _i_  3  2  -U  9 

12.  Given  the  function:  /(z)  =  —     _  .    Does  this  function  converge 

uniformly  to  its  values  along  the  circle  C  having  the  origin  as  a  center  and  a 
radius  equal  to  one?  Does  the  Cauchy-Goursat  theorem  apply  to  the  integral 
taken  along  C?  to  a  circle  concentric  with  C  but  lying  within  C? 

13.  Givenw=f{z)=3z^  +  7z  +  4,      z  =  <t>{r)  =  ^  ""'  "t  ^.     Ibw  ==f  ]<!>(?■)  ] 

T  -f-  1 

holomorphic  for  values  of  |  t  |  <  1? 

14.  Given  the  function  /(«)  defined  by  the  relation 


I  —  z  ' 


where  t  takes  complex  values  along  the  circle  C  of  radius  2  about  the  origin. 
Compute  the  values  of /(«)  and/"(z)  f or  z  =  1  +  t.  -    ' 

16.  Given  the  function  /(z)  =3z'  +  4z*  +  7zH-2.  Find  the  integral  of 
this  function  along  the  circle  x^  +  y^  =  1  from  the  point  a  =  (1,  0)  to  the  point 
/3  =  (  —  1,0).  Show  that  this  integral  taken  around  the  complete  circle  is  im- 
changed  when  any  regular  closed  curve  is  substituted  for  this  circle  as  the  path 
of  integration. 

'   16.   Given  u  {x,y)  =  ar*  —  6  xh/^  +  y*.     Find  a  function  v  (x,  y)  such  that 
ti  H-  if  is  an  analytic  function /(z).     Find  the  value  of /"(z)  for  z  =  2  +  3 1. 

17.  Given  any  rational  integral  function  /(z).  Show  how  the  value  of  J{z) 
f or  z  =  2  4-  3  t  can  be  found  when  we  know  the  values  of  /(z)  on  the  circle  about 
the  origin  having  a  radius  p  =  4. 

18.  An  incompressible  fluid  flows  over  a  plane  with  a  velocity-potential 

u  =  x"^  —  y^. 

Determine  a  value  of  v  such  that 

w  =  u  +iv  =  f(z), 

is  holomorphic  in  the  finite  region.     Find  the  components  of  the  velocity  and  the 
direction  of  the  flow  at  the  point  z  =  3  +  2i. 


^ 


■^1 


V ',  9-  vy 


I  //  j,^a  <:^r-t 


CHAPTER  IV  ?'i-5'7 


MAPPING,  WITH  APPLICATIONS  TO   ELEMENTARY 
FUNCTIONS 

26.  Conjugate  functions.  In  the  present  chapter  we  shall  dis- 
cuss certain  elementary  functions  with  special  reference  to  the 
correspondence  between  certain  portions  of  the  Z-plane  and  the 
"PF-plane,  as  determined  by  the  relation  between  the  function  w  and 
the  independent  variable  z.  Before  taking  up  this  general  discussion, 
however,  we  shall  consider  the  significance  of  u  and  v  in  the  relation 

w  =  f{z)  =  u{x,  y)  +  w{x,  y),  (1) 

where  j{z)  is  holomorphic  in  a  given  region.  As  we  have  seen,  both 
u  and  V  satisfy  Laplace's  differential  equation  and  hence  either  may 
be  considered  as  a  potential  function.  They  are  consequently  of  im- 
portance in  theoretical  physics.  Because  of  the  relation  that  each 
function  has  to  the  other,  they  are  called  conjugate  functions. 
We  can  represent  the  functions  u{x,  y),  v{x,  y)  by  the  two  surfaces 

u  =  u{x,  y),        V  =  v{x,  y), 

where  x,  y,  u  and  x,  y,  v  are  two  systems  of  Cartesian  coordinates  of 
points  in  space.     These  two  surfaces  represent  then  the  real  and 
the  imaginary  parts  of  the  given  function  w  =  f(z). 
Consider,  for  example,  the  function 

w  =  z^. 
We  have  then 

w  =  u  +  iv  =  {x  -\-  iyY  =  x^  -\-  2iy  —  y^. 

By  equating  the  real  and  the  imaginary  parts,  we  get 
u  =  x^  —  y^,        V  =  2  xy. 

Each  of  the  surfaces  representing  these  equations  is  cut  by  any 
plane  parallel  to  the  XF-plane  in  a  rectangular  hyperbola.  From 
the  w-surface,  we  get  a  system  of  such  hyperbolas  having  the  lines 
y  =  ±  X  as  the  asymptotes.  From  the  w-surface,  we  obtain  a 
system  of  rectangular  hyperbolas  having  the  two  axes  as  asymptotes. 

101 


^'..f 


'\ 


102 


MAPPING,  ELEMENTARY  FUNCTIONS 


[Chap.  IV. 


The  two  systems  of  curves  when  projected  upon  the  XF-plane  appear 
as  in  Figs.  33,  34,  respectively.  In  either  case  the  curves  are  the 
projections  of  the  intersections  of  the  given  surfaces  with  a  system 
of  equiangular  hyperbohc  cylinders  whose  generating  lines  are  per- 
pendicular to  the  Xy-plane. 

We  have  considered  the  two  surfaces  u  =  u(x,  y),  v  =  v{x,  y)  as 
related  to  distinct  coordinate  axes.    Suppose  now  we  think  of  them 


Fig.  33. 


Fig.  34. 


as  being  referred  to  the  same  system  of  axes.  The  projection  upon 
the  XF-plane  of  th6  curves  of  intersection  reveals  an  important 
relation  between  the  two  systems  of  curves.  It  will  be  shown  that 
these  two  systems  of  curves,  given  by 


U  =  Ci, 


V  =  C2, 


where  Ci,  Cj  are  constants,  are  orthogonal  systems  in  the  XF-plane. 
To  do  this,  we  make  use  of  the  slope  of  the  curves,  which  is  given  by 

^  •    From  the  relation  w(x,  y)  =  Ci,  we  have  for  —  5^  0 

In  order  that  the  curves  given  by  v(x,  y)  =  c^he  orthogonal  to  the 
system  u(x,  y)  =  ci,  the  slope  of  v(x,  y)  =  d,  at  points  of  intersec- 
tion with  the  curves  u{x,  y)  =  Ci  must  be  the  negative  reciprocal  of 


dy 
dx 

dx 
du 

dy 

Art.  26.] 


CONJUGATE  FUNCTIONS 


103 


the  slope  of  u{x,  y)  =  Ci  at  these  points.     The  general  condition  of 
orthogonality  is  then 

dv      du 

dx  _  dy 

dv      du' 

dy      dx 

which  may  be  written  in  the  form 

du     dv      du     dv  _ 
dy     dy      dx     dx 

This  condition  is  satisfied  by  conjugate  functions;  for,  we  know 
that  the  conjugate  functions  u{x,  y)  and  v{x,  y)  satisfy  the  Cauchy- 
Riemann  differential  equations 

du  _  dv       du  _       dv 

dx      dy^     dy  dx 

Multiplying  these  two  equations  member  by  member,  we  have  pre- 
cisely the  condition  of  orthogonality  given  above. 

The  fact  that  these  two  systems  of  curves  are  orthogonal  increases 
the  ease  with  which  either  curve  may  be  constructed  when  the  other 
is  given.  AU  we  need  to  do  is  to 
construct  a  second  system  every- 
where orthogonal  to  the  first. 
When  so  drawn,  the  two  sets  of 
curves  obtained  in  the  foregoing 
example  are  as  shown  in  Fig.  35. 
If  we  think  of  both  systems  of 
curves  as  projected  back  upon 
each  of  the  surfaces  u  =  u(x,  y), 
v  =  v{x,  y),  we  shall  have  upon 
each  surface  two  systems  of  curves 
cutting  each  other  at  right  angles. 

The  curves  cut  from  either 
surface  by  planes  parallel  to  the 
Xy-plane  are  called  the  level  lines.  The  curves  of  the  orthogonal 
system  are  called  the  lines  of  slope,  or  the  curves  of  quickest  descent. 
In  theoretical  physics  other  special  names  are  employed  to  designate 
the  projection  of  these  two  systems  upon  the  XF-plane. 

In  an  electric  field  or  in  the  field  of  a  gravitational  force,  a  surface 
such  that  the  potential  is  the  same  at  all  points  of  it  is  called  an 
equipotential  surface.     Hence,  any  right  cylinder  through  the  lines 


Fig.  35. 


104  MAPPING,   ELEMENTARY  FUNCTIONS  [Chap.  IV. 

of  level  on  the  w-surface  or  the  y-surface  just  discussed  is  an  equi- 
potential  surface.  Since  one  such  surface  may  pass  through  each  level 
line,  we  have  a  system  of  equipotential  surfaces.  Curves  drawn  per- 
pendicular to  these  surfaces  are  called  lines  of  force. 

The  traces  of  the  equij)otential  surfaces  upon  the  XF-plane  are 
called  equipotential  lines.  In  the  applications  to  be  considered,  the 
lines  of  force  as  well  as  the  equipotential  lines  lie  in  a  plane,  which 
we  shall  take  as  the  complex  plane.  In  the  case  of  the  flow  of  an 
incompressible  fluid  or  of  streaming  in  electricity  the  two  orthogonal 
systems  of  curves  are  referred  to  as  the  equipotential  lines  and  the 
lines  of  flow  respectively;  while  in  the  theory  of  heat  they  are  called 
the  isothermal  lines  and  the  lines  of  flow.  In  the  case  of  electric 
currents  the  lines  of  flow  are  frequently  called  stream-lines. 

We  shall  have  frequent  occasion  in  this  chapter  to  return  to  the 
properties  of  conjugate  functions. 

27.  Conformal  mapping.  The  relation  w  =  f{z)  gives  a  definite 
association  between  those  points  of  the  complex  plane  representing 
the  values  of  z  and  those  representing  the  values  of  w.  As  a  matter 
of  convenience  it  is  usual  to  represent  the  ^-points  in  one  plane, 
called  the  Z-plane,  and  the  w-points  in  another  plane,  called  the 
TT-plane.  These  two  planes  have  a  relation  to  each  other  somewhat 
similar  to  that  which  the  two  coordinate  axes  have  in  the  considera- 
tion of  functions  of  a  real  variable.  As  the  point  P  traces  any  curve 
in  the  Z-plane,  the  corresponding  point  Q  will  trace  a  curve  in  the 
TF-plane.  We  express  the  relation  between  the  two  curves  by  say- 
ing that  the  curve  in  the  Z-plane  is  mapped  upon  the  TF-plane.  In 
discussing  the  general  properties  of  mapping,  it  is  often  convenient 
to  speak  of  the  mapping  of  the  one  plane  upon  the  other^rather  than 
of  the  mapping  of  some  particular  configuration  from  the  on«  plane 
upon  the  other.  If  J{z)  is  multiple-valued,  then  to  each  point  in 
the  Z-plane  there  correspond  in  general  several  distinct  points  in 
the  W-plane.  In  such  cases  it  is  often  convenient  to  map  the  whole 
of  the  one  plane  upon  a  portion  of  the  other. 

From  the  discussion  in  Art.  22,  we  are  able  to  state  the  conditions 
under  which  the  W-plane  can  be  mapped  in  a  definite  manner  upon 
the  Z-plane.     We  have  seen  that  if  the  Jacobian 


du  du 

dx  dy 

dv  dv 

dx  dy 


/  :)M 


Art.  27.]  CONFORMAL   MAPPING  105 

does  not  vanish  within  a  given  region  of  the  Z-plane,  which  is  the 
case  if  f'{z)  9^  0,  we  can  always  solve  the  equations 

u  =  ^i(x,  y), 

V  =  ^2(.x,  y)  (1) 

for  X,  y  m.  terms  of  u  and  v.  Moreover,  there  is  but  one  such  solu- 
tion possible.     Denoting  the  result  of  this  solution  by 

X  =  xi(w,  v), 

y  =  X2(w,  v),  (2> 

we  can  by  means  of  these  equations  map  in  a  definite  manner  the 
TF-plane  upon  the  Z-plane.  Whether  the  W-plane  maps  upon  the 
entire  Z-plane  or  only  upon  a  portion  of  it  depends  in  general  upon 
the  character  of  the  two  functions  xi,  X2.  By  means  of  relations  (1)^ 
(2),  we  can,  however,  establish  a  one-to-one  correspondence  between 
the  points  of  a  region  of  the  Z-plane  and  those  of  a  corresponding 
region  of  the  W-plane;  that  is  to  say,  if  ^T  is  the  region  of  the 
Z-plane  under  consideration  and  S  the  corresponding  region  of  the 
TF-plane,  then  to  each  point  of  T  there  corresponds  one  and  only 
one  point  of  S  and  conversely. 

Ex.  1.     Given  w  =  z^.     Let  it  be  required  to  map  a  given  configuration  from 
the  Z-plane  upon  the  TF-plane  and  conversely  by  means  of  this  relation. 

As  in  Art.  25,  we  have 

u  =  x'^  —  y^,        V  =  2  xy. 
We  may  also  write 

z  =  p(cos  0  +  i  sin  6), 
w  =  p'(cos  0'  +i  sin  0')  =  p^(cos  2  d  +  i  sin  2  d). 

Hence,  we  have 

p'  =  p2,         e'  =  2  0. 

From  the  relation  between  0  and  0',  it  will  be  seen  that  a  half  of  the  Z-plane  maps 
into  the  whole  of  the  PF-plane,  and  on  the  other  hand  a  half  of  the  W-plane  maps 
into  a  quadrant  of  the  Z-plane;  for  example,  the  upper  half  of  the  TT-plane  maps 
into  the  first  quadrant  of  the  Z-plane. 

To  map  from  the  TT-plane  upon  the  Z-plane,  suppose  we  put  u  =  c,  a  constant. 
We  obtain  a  rectangular  hyperbola  given  by  the  equation 


Regarding  c  as  a  variable  parameter,  we  have  two  systems  of  rectangular  hyper- 
bolas having  respectively  the  lines  y  =  zLx  as  asjTnptotes,  according  as  c  is  posi- 
tive or  negative.  As  the  upper  half  of  the  W-plane  maps  into  the  first  quadrant 
of  the  Z-plane,  the  given  lines  u  =  c  map  into  those  branches  of  these  hyperbolas 


106 


MAPPING,  ELEMENTARY  FUNCTIONS 


[Chap.  IV. 


situated  in  that  quadrant,  as  represented  in  Fig.  37.  For  v  =  c',  a,  positive  con- 
stant, we  obtain  in  the  Z-plane  a  system  of  hyperbolas  orthogonal  to  the  first 
systems.  We  see  then  that  the  upper  half  of  the  W-pl&ne  maps  into  the  first 
•quadrant  of  the  Z-plane  and  that  the  orthogonal  systems  of  lines  parallel  to  the 


FiQ.  36. 


Fig.  37. 


two  axes  of  the  TF-plane  map  into  orthogonal  systems  of  rectangular  hyperbolas, 
having  respectively  the  two  positive  axes  and  the  line  y  =  x  as  limiting  cases. 

Let  us  now  undertake  to  map  certain  simple  curves  of  the  Z-plane  upon  the 
TF-plane.    We  know  from  what  has  been  said  that  a  half  of  the  Z-plane  will  map 


Fig.  38. 


Fig.  39. 


into  the  whole  of  the  TT-plane.     Consider  the  line  x  =  0  of  the  Z-plane.     We 
have  in  the  W-plane 

w  =  -j/2; 

that  is,  for  any  value  of  y,  either  positive  or  negative,  w  has  a  negative  real  value. 
Consequently,  the  whole  of  the  F-axis  maps  into  the  negative  [/-axis.     The 


Art.  27.] 


CONFORMAL  MAPPING 


107 


points  Bi,  B2  (Fig.  39),  map  into  the  same  point  Q  in  the  TT-plane  (Fig.  38).  If 
2  describes  a  semicircle  of  radius  a  as  indicated,  then  w  describes  a  complete  circle 
of  radius  a^  about  the  origin  O'.  If  z  describes  the  line  x  =  c,  then  w  describes 
a  parabola  cutting  the  C/-axis  at  c^;  for,  eliminating  y  between  the  equations 

u  =  d^  —  y^,         V  =  2  cy, 
we  have  v*  =  40^(0^  —  u). 

The  position  of  this  parabola  is  shown  in  Fig.  38.  In  a  similar  manner  any  other 
curve  in  the  Z-plane  may  be  mapped  upon  the  TF-plane. 

The  given  function  determines  an  electrostatic  field  *  in  the  immediate  vicin- 
ity of  two  conducting  planes  at  right  angles  to  each  other.  In  this  field  the 
equipotential  surfaces  are  the  system  of  hyperbolic  cylinders  determined  by  the 
equation  v  =  2xy.  As  a  special  case  we  have  the  two  planes  intersecting  at  right 
angles.  The  relation  between  w  and  z  also  determines  the  field  between  two  coaxial 
rectangular  hyperbolas.  The  system  of  hyperbolas  v  =  c  are  the  lines  of  equi- 
potential, while  the  curves  of  the  orthogonal  system  u  =  c  are  the  lines  of  force. 

Let  us  now  consider  the  general  case  where  one  region  of  the  com- 
plex plane  is  mapped  upon  another  by  means  of  a  function  w  =  f(z) 
which  is  holomorphic  for  the  values  of  z  under  consideration.  We 
shall  inquire  into  the  effect  of  such  mapping  upon  the  angle  that 
one  curve  makes  with  another  at  their  point  of  intersection.  If  the 
magnitude  of  the  angle  is  preserved,  even  though  reversed  in  direc- 
tion the  mapping  is  said  to  be  isogonal  or  conformal.  We  shall 
now  demonstrate  the  following  proposition.  

Theorem.  The  mapping  of  the  Z-plane  upon  the  W-plane  hy 
means  of  a  function  w  =  f{z)  is  isogonal,  without  reversion  of  angles, 
in  the  neighborhood  of  a  regular  point  Zo  of  f{z),  provided  f  {zq)  9^  0. 


a 


u 


o 


Fig.  40. 


Fig.  41. 


Let  Ci,  Ci  be  any  two  curves  in  the  Z-plane  intersecting  at  Zq. 
Suppose  Ci,  C2  map  into  the  two  curves  *Si,  ;S2  of  the  TF-plane  inter- 
secting in  the  point  Wq  corresponding  to  Zq.    We  are  to  show  that  the 
*  See  Jeans,  Electricity  and  Magnetism,  p.  262. 


108  MAPPING,  ELEMENTARY  FUNCTIONS  [Chap.  IV. 

angle  that  Ci  makes  with  d  is  the  same  as  that  which  aSi  makes 
with  St.    The  relation  between  w  and  z  is  given  by  the  function 

w=f{z). 

Since  Zq'is  a.  regular  point,  the  derivative  of  f{z)  exists  and  is  defined 
by  the  relation 

rw  =  t  f  •  (3) 

Az=0    i^Z 

Since  f'(zo),  Aw,  Az  are  all  complex  numbers  and  /'(zo)  5^  0,  we  may 
write 

f'{zo)  =  p(cos  d  -{-isind),  p  9^  0, 

Aw  =  pi(cos  di  +  i  sin  ^i),      Az  =  p2(cos  02  +  i  sin  ^2),  (4) 

where  di,  dz  are  taken  to  be  the  chief  amplitudes  of  Aw,  Az,  respec- 
tively. 
From  (3)  it  follows  that 


Pi 


f'(zo)  =    L  -  (cos  di-di-hisindi-  62) 

Ar=0  P2 


Pi 


=    L  -    L  (cos 01  -  02  +  ^sin0^  -  ^2),       '  (5) 

Az=0  P2Az=0 

=    L  -  }  COS  L  (01  -  ^2)  +  i  sin  L  {di  -62)  I, 

since  cos  z  and  sin  z  are  continuous  functions. 
We  have  therefore 

p=    L^-^,     6=    L  (6,-62).  (6) 

Az=0  P2  Az=0 

Denote  by  <^  the  angle  that  the  tangent  to  the  curve  Ci  at  Zq 
makes  with  the  positive  X-axis  and  by  ^1  the  angle  that  the  tangent 
to  the  corresponding  curve  *Si  makes  with  the  positive  17-axis.  We 
have  then,  since  Aw  approaches  zero  with  Az, 

L  62  =  <j>2)        L  61  —  <f>i.  (7) 

Az=0  A2=0 

The  amplitude  of  /'(zo)  is  less  numerically  than  2  t,  because  of  the 
restriction  of  the  values  of  di,  62  to  the  chief  amplitudes  of  Aw,  Az. 
We  may  then  write  ^f\ 

6  —  <i>i  —  <h'i  (8) 

that  is  to  say,  6  represents  the  angle  through  which  the  curve  Ci  is 
turned  in  the  process  of  mapping.     As  f'(zo)  is  a  constant,  6  is  also  a 


Akt.  27.]  CONFORMAL  MAPPING  109 

constant  for  all  curves  passing  through  Zo;  that  is,  every  such  curve 
is  turned  through  the  same  angle  6.      Hence  Si  makes  the  same 
angle  with  Si  as  Ci  makes  with  d.     The  mapping  is  therefore  not 
only  isogonal  but  the  direction  of  the  angle  is  not  reversed. 
From  (6)  we  have 


Pi 

I  =    ^ 

which  may  be  written  in  the  form 


^  =  p4-6,  (9) 

where  c  vanishes  with  Az.  Hence,  the  ratio  of  the  magnitude  of 
an  element  of  the  resulting  configuration  to  the  magnitude  of  the 
corresponding  element  in  the  given  configuration  is  approximately 
P  =  I  f'(^)  I  >  which  may  be  called  the  ratio  of  magnification  in  the 
neighborhood  of  Zq.  The  approximation  is  closer  the  smaller  the 
element.  Since  p  is  constant  for  Zo  we  conclude  that  the  similarity 
of  infinitesimal  elements  is  preserved. 

If  instead  of  mapping  the  two  given  curves  Ci,  0%  by  means  of 
the  function 

w  =  /(z)  =  u  -^  iv,  (10) 

the  mapping  had  been  done  by  means  of  the  relation 

wi  =fi{z)  =  u-  iv,  (11) 

the  resulting  configuration  would  have  been  situated  below  the  C/-axis 
as  shown  in  Fig.  42.  The  configurations  obtained  by  (10)  and  (11) 
are  symmetrical  to  each  other  with  respect  to.  the  CZ-axis.  We  say 
that  by  this  change  the  resulting  configuration  has  been  reflected 
upon  the  ?7-axis.  It  will  be  observed  that  in  mapping  by  means 
of  the  function  given  in  (11)  the  direction  of  the  angle  that  the  one 
curve  makes  with  the  other  has  been  reversed  in  the  resulting  con- 
figuration. The  mapping  by  (11)  may  be  described  as  isogonal  but 
with  reversion  of  angles. 

Mapping  by  means  of  a  function  which  is  holomorphic  in  a  given 
region  is  but  a  special  case  of  conformal  mapping.  One  surface  may 
be,  in  fact,  mapped  conformally  upon  another  if  we  have  the  relation 

ds  =  M  dS, 

where  ds,  dS  are  elements  of  arcs  taken  in  any  direction  from  corre- 
sponding points  upon  the  two  surfaces  and  M,  the  ratio  of  magni-r 


110 


MAPPING,  ELEMENTARY    FUNCTIONS 


[Chap.  IV. 


fication,  depends  upon  the  variable  coordinates  but  is  independent 
of  the  differential  elements.*  In  the  special  case  considered  in  the 
theorem  the  general  factor  M  is  replaced  by  |  f'(zo)  \  . 


O' 


ri 


,'Sa 


•St 


u 


o 


Fig.  42. 


Fig.  43. 


At  those  points  of  the  complex  plane  where /'(z)  =  0,  the  mapping 
may  cease  to  be  conformal,  even  if  the  given  points  are  regular  points 
of  the  function  w  =  f{z).  For  example,  consider  the  function  w  =  z^ 
in  the  neighborhood  of  the  point  2  =  0.     Putting 

f{z)  =  p(cosd  -\-isme), 
z  =  p'(cos  d'  +  i  sin  6'), 
we  have,  since 

f'(z)=D.(z^)=^2z, 
p(cos d  -\-isine)  =  2 p'(cos d'  +  i sin d') . 

But  as  p  =  p'  =  0  for  z  =  0,  this  relation  has  no  significance.  As  a 
matter  of  fact,  as  we  have  already  seen  (Ex.  1),  any  two  curves 
intersecting  in  the  origin  at  a  given  angle  map  by  means  of  the  func- 
tion w  =  z^  into  two  curves  intersecting  at  an  angle  of  twice  that 
magnitude.     Consequently,  it  can  not  be  asserted  that  the  mapping 

*  See  Schefifere,  Anwendung  der  Differential  und  Integralrechnung  auf  Geo- 
melrie,  Vol.  II,  p.  70;  also  Osgood,  Lehrbuch  der  Funktionentheorie,  2<1  Ed., 
p.  79  et  seq. 


Art.  27.]  CONFORMAL  MAPPING  111 

by  means  of  the  functional  relation  w  =  z^  is  isogonal  in  the  neigh- 
borhood of  the  origin. 

From  what  has  been  said  concerning  isogonality,  it  must  not  be 
inferred  that  the  map  in  the  TF-plane  is  as  a  whole  identical  or  even 
similar  to  the  original  configuration  in  the  Z-plane.  The  amount 
of  distortion  that  takes  place  depends  upon  the  coordinates  of  the 
point.  To  show  this,  consider  the  value  of  the  ratio  of  magnifica- 
tion p  =  \f'{z)  |.    We  have 


p  =  \riz)\  = 


du   ,    .dv  \       \dv    ,    .dv 
dx         dx\        dy        dx 


=vlHiJ=v/.^: 


dv       du     dv 
dy      dy     dx 


It  is  evident  from  this  relation  that  p  depends  upon  the  variables 
X,  y,  and  therefore  may  change  with  the  point  z.  The  functional 
relation  between  w  and  z  does  not,  therefore,  necessarily  establish  a 
similarity  between  finite  parts  of  the  two  corresponding  configu- 
rations. 

The  geometric  significance  of  the  derivative  may  be  regarded  as 
a  generalization  of  the  significance  of  the  derivative  in  the  calculus 
of  real  variables;  for,  let  Zo  be  any  given  point  in  the  Z-plane  and 
Wo  its  image  in  the  W-plane.  As  z  passes  through  the  point  Zo  in  any 
direction  w  passes  through  Wq  ia  a,  corresponding  direction.  The 
modulus  p  of  f{zo)  gives  the  limit  of  the  ratio  of  the  absolute  value 
of  the  change  that  takes  place  in  w  to  the  absolute  value  of  the 
change  that  takes  place  in  z;  that  is,  p  measures  the  magnification 
about  the  point  Wo  of  the  infinitesimal  elements  of  the  configuration 
in  the  TT-plane  relative  to  the  corresponding  infinitesimal  elements 
of  the  Z-plane.  This  change  corresponds  in  the  calculus  of  a  real 
variable  to  the  change  in  the  ordinate  y  as  x  varies,  determining  in 
that  case  the  slope  of  the  tangent  to  the  curve.  On  the  other  hand, 
the  amplitude  6  of  f{zo)  gives  the  amount  of  rotation  between  c5r- 
responding  elements  of  the  two  planes.  Both  p  and  6  may  change 
with  z  since  f'{z)  is  in  general  a  function  of  z. 

Further  interpretations  of  the  derivatives  are  frequently  made  in 
solving  physical  problems.  If  we  let  z  move  along  a  definite  curve, 
then  w  likewise  moves  along  some  curve  in  the  W-plane.  From  the 
relation  between  w  and  z  we  have 

dw=J'{z)dz.  (12) 


112  MAPPING,  ELEMENTARY  FUNCTIONS  [Chap.  IV. 

Considering  the  time  t  in  which  the  motion  takes  place  as  the  com- 
mon variable  in  terms  of  which  the  changes  of  w  and  z  are  expressed, 
we  may  replace  the  differentials  in  (12)  by  time  derivatives  and 

D,w=f'(2)DtZ,  (13) 

whence  \Dtw\  =  \f{z)  \'\Dt2\. 

The  derivatives  Dtw,  DtZ  represent  the  velocities  both  as  to  magni- 
tude and  direction  with  which  the  points  w  and  z  move  along  their 
respective  curves.  The  speeds  with  which  these  motions  take  place 
are  given  by  |  DtW  | ,  |  DtZ  \  respectively.     The  derivative 

fiz)  =  p(cosd-\-i  sin  d)  (14) 

then  gives  the  ratio  of  the  two  velocities,  while 

P=l/'(^)1  (15) 

gives  the  ratio  of  the  speed  of  tt;  to  that  of  z. 

By  means  of  the  second  time  derivatives  of  w  and  z  the  accelera- 
tion of  the  moving  point  may  be  determined  at  any  instant.  Differ- 
entiating (13)  we  have  7 

Dtho  =  f'{z)  .  {Day  +  f(z)  .  Dt'z.  (16) 

The  modulus  of  Dtho  gives  the  magnitude  of  the  acceleration  and 
the  ampUtude  of  Dthu  gives  the  direction  in  which  the  acceleration 
takes  place.  Both  the  magnitude  and-  the  direction  of  the  accelera- 
tion of  the  ly-point  involve  the  velocity  as  well  as  the  acceleration 
of  the  corresponding  z-point  since  Dt^w  depends  upon  both  DtZ 
and  Dt^z.  The  following  example  illustrates  the  questions  under 
discussion. 

Ex.  2.  Given  w  =  z*.  Let  z  start  from  the  point  i  with  an  initial  velocity 
of  one  centimeter  per  second  and  move  with  uniform  velocity  along  a  line  parallel 
to  the  positive  axis  of  reals.  Determine  the  path  of  the  corresponding  i^point 
and  the  velocity,  acceleration,  and  speed  of  that  point  at  any  time  t. 

The  path  of  the  wvpoint  is  the  map  upon  the  TT-plane  of  the  positive  half  of 
the  line  y  =  I.     This  line  maps  into  the  upper  half  of  the  parabola  v^  =  4u  -f-  i. 

The  velocity  v  of  the  t^vpoint  is  given  by  the  relation 

dz  . 
The  derivative  ^  is  the  velocity  of  the  z-point,  which  in  this  case  is  constant  and 

equal  to  one  centimeter  per  second.     Hence,  we  have 

„  =  2a- 1  =  22. 


M 


Abt.  27.1  CONFORMAL  MAPPING 

The  acceleration  a  is  given  by  the  relation 


113 


+fwf-^. 


Consequently,  as  the  z-point  starts  at  z  =  i  and  moves  as  given  by  the  con- 
ditions stated  in  the  problem,  w  starts  at  the  point  w  —  i^  =  —  1  and  moves 
along  the  upper  half  of  the  parabola,  starting  with  an  initial  velocity  of 

„  =  22  =  2i. 

At  the  end  of  any  given  time,  say  3  seconds,  we  have 

2  =  zo  =  3  +  i,         tTo  =  2o*  =  8  +  6 1, 

and  vo  =  2zo  =  6  +  2t. 

The  acceleration  of  the  ip-point  remains  constantly  equal  to  two  centimeters 
per  second  per  second.     The  acceleration  in  this  case  being  a  real  number  its 


Y 


O 


Fig.  44. 


Fig.  45. 


amplitude  is  zero  and  it  is  directed  at  each  point  parallel  to  the  positive  [/-axis 
as  shown  in  Fig.  44. 

The  direction  of  the  velocity  at  any  point  is  determined  by  the  amplitude  of 
2,  since  v  =  2z.  As  the  velocity  is  always  measured  along  the  tangent  to  the 
path  of  the  moving  point,  it  foUows  that  the  tangent  to  the  t»-curve  is  always 
parallel  to  the  half-ray  from  the  origin  to  the  point  z. 

The  speed  of  the  t/j-point  at  any  instant  is 

\Dtw\  =  \r(.z)\'\Dtz\=2-\z\. 
At  the  end  of  3  seconds  the  speed  is  then 


2-|zo|=2\/9  +  l=2  VlO 


114  MAPPING,   ELEMENTARY  FUNCTIONS  [Chap.  IV. 

28.  The  function  w  =  «".  In  a  previous  article  we  have  dis- 
cussed a  special  case  of  the  general  function  w  =  2",  namely,  the  case 
where  n  =  2.  There  are  some  additional  properties  of  the  general 
case  that  will  now  be  considered. 

The  function  ty  =  2;^  is  the  simplest  case  that  we  have  of  a  general 
class  of  functions  known  as  linear  automorphic  functions.  By  such 
a  function  we  mean  one  that  remains  unchanged  when  the  inde- 
pendent variable  is  replaced  by  any  one  of  a  definite  set  of  its  linear 
substitutions  such  that  the  set  form  a  group.*  In  this  case  the  func- 
tion remains  unchanged  under  the  linear  substitutions  consisting  of 
z  =  —  2'  and  the  identical  substitution  z  =  z'. 

It  may  be  shown  that  the  general  function  w  =  z"  is  likewise  an 
automorphic  function.  To  find  the  particular  linear  transforma- 
tions that  leave  the  function  unchanged,  we  make  use  of  the  number 

27r    ,    .    .    27r 

0)  =  cos H  I  sm 

.n  n 

This  number  is  one  of  the  n^^  roots  of  unity;   for,  by  DeMoivre's 
theorem,  we  have 


j"  =    cos  — 


.  .    2t\ 

-f- 1  sm  —     =  cos  2  TT  +  t  sm  2  TT  =  1. 
n 


The  other  n***  roots,  aside  from  unity  itself,  are 

cd^,  oj^,   .   .   .  a;"~^ 

Moreover,  since 

(CO*)"  =  1,  (A;  =  1,  2,  3,  .  .  .  ,  n  -  1), 
we  may  write 

(w*  •  2)"  =  2". 

Hence,  if  co*  2',  where  /b  =  0,  1,  2,  3,  .  .  .  ,  n  —  1,  is  substituted  for 
2  in  the  function  w  =  2",  the  function  remains  unchanged.  The  sub- 
stitutions 2  =  a}*2'  form  a  group  of  linear  substitutions  and  hence 
«;  =  2"  is  therefore  a  linear  automorphic  function. 

In  the  discussion  of  the  function  w  =  2^,  attention  was  called  to 
the  fact  that  a  half  of  the  Z-plane  maps  into  the  whole  of  the 
TT-plane.    Let  us  now  consider  the  general  case,  where  w  =  2".    We 

^^®  z  =  p(cosd  +  isin9), 

w  =  p'(cos  6'  -\-  i  sin  6'). 

*  See  Fricke,  Enq^klopddie  d.  Math.  Wiss.,  Vol.  II2,  p.  351. 


Art.  28.]  THE  FUNCTION  Z"  115 

From  these  relations  we  obtain 

p  =  p",     d'  =  nd. 

Here,  as  in  the  special  case  where  w  =  2,  a  circle  about  the  origin  in 
the  Z-plane  maps  into  a  circle  in  the  W-plane,  and  a  straight  line 
through  the  origin  becomes  a  straight  line  through  the  origin  in  the 

TF-plane.    From  the  relation  between  6  and  6',  it  will  be  seen  that  I  -  j 

of  the  circle  in  the  Z-plane  maps  into  the  whole  of  the  circle  in  the 

TF-plane;  consequently,  a  sector  bounded  by  any  two  haK-rays  drawn 

27r 
from  the  origin  making  an  angle  —  radians  with  each  other  maps 

into  the  whole  of  the  TF-plane.  The  values  of  d'  corresponding  to  the 
chief  amplitude  of  w  belong  to  the  interval 

—ir<d'=  TT. 

The  sector  bounded  by  ORi,  OR2  (Fig.  47),  making  respectively  the 

TTTT.  ••  •  •  .  • 

angles  - , with  the  positive  X-axis,  maps  in  a  continuous,  smgle- 


'//////\////////W///////////M///mM///////M. 


O' 


Fig.  46. 


-^U 


Fig.  47. 


^X 


valued  manner  upon  the  entire  TT-plane.  The  lower  bank  of  the 
line  ORi  goes  over  into  the  upper  bank  of  the  negative  axis  of  reals 
of  the  TT-plane.  The  upper  bank  of  the  line  OR2  maps  into  the  lower 
bank  of  the  negative  axis  of  reals  of  the  TT-plane  as  shown  in  Figs. 
46  and  47. 


116 


MAPPING,  ELEMENTARY  FUNCTIONS 


[Chap.  IV. 


Any  portion  of  the  Z-plane  which  maps  into  the  entire  TF-plane 

is  called  a  fundamental  region  of  the  given  function.     Thus  the 

region   R1OR2  is  a   fundamental  region  of    the  function  w  =  z". 

Likewise  any  sector  bounded  by  two  half-rays  from  the  origin  and 

2ir 
making  an  angle  of  —  with  each  other  may  be  taken  as  a  funda- 
n 

mental  region.  In  the  case  of  linear  automorphic  functions,  any  sub- 
stitution that  leaves  the  function  unchanged  maps  any  fundamental 
region  of  the  function  into  another  region  that  does  not  overlap  the 
first.  The  second  region  may  be  taken  likewise  as  a  fundamental 
region  of  the  given  function. 

It  is  to  be  observed  that  while  the  mapping  of  the  Z-plane  upoii 
the  TT-plane  by  means  of  the  given  function  is  continuous  and 
single-valued,  it  does  not  follow  that  the  mapping  of  the  TF-plane 
back  upon  the  fundamental  region  of  the  Z-plane  is  continuous. 
As  a  matter  of  fact,  such  a  mapping  is  single-valued  but  not  con- 
tinuous. In  other  words,  not  every  continuous  curve  in  the  TF-plane 
maps  into  a  continuous  curve  in  the  fundamental  region  of  the 
Z-plane.  Suppose,  for  example,  that  we  have  a  curve  in  the  TF-plane 
crossing  the  negative  axis  of  reals.  As  we  have  already  seen,  the 
upper  bank  of  the  negative  C7-axis  maps  into  the  lower  bank  of  the 
line  ORi;  while,  on  the  other  hand,  the  lower  bank  of  the  nega- 
tive C/-axis  maps  into  the  upper 
bank  of  the  line  OR2.  Conse- 
quently, the  curve  C  (Fig.  46), 
which  is  a  continuous  curve  in  the 
TF-plane,  becomes  a  discontinuous 
curve  in  the  Z-plane.  We  can  say, 
however,  that  the  mapping  from 
the  TF-plane  to  the  fundamental 
region  of  the  Z-plane  is  a  single- 
valued  process;  for,  to  every  point 
-^  of  the  TF-plane  there  is  one  and 
Fio,  48,  only  one  corresponding  point  in  the 

fundamental  region  of  the  Z-plane. 
A  fundamental  region  of  the  function  wj  =  z"  need  not  be  bounded 
by  straight  Unes.     It  serves  our  purpose  equally  well  to  take  any  con- 
tinuous curve  proceeding  from  the  origin  and  to  revolve  it  through 

27r 
the  angle  — ,  as  indicated  in  Fig.  48.     The  boundary  lines  of  the 


Art.  28.]  THE  FUNCTION   2"  117 

sector  extend  indefinitely  from  the  origin  in  the  case  of  the  function 
under  discussion. 

It  must  not  be  assumed  that  the  fundamental  region  of  all  auto- 
morphic  functions  can  be  obtained  in  this  manner.  The  function 
here  considered  is  a  special  case.  A  more  general  case  will  be  dis- 
cussed in  a  subsequent  chapter,  in  connection  with  doubly  periodic 
functions. 

By  means  of  the  fundamental  operations  of  multiplication  and 
addition  applied  to  complex  constants  and  functions  of  the  type 
w  =  2",  where  n  is  integral,  we  obtain  the  rational  integral  function 

f{z)  =  002"  +  q:i2"-^  +  a2Z"~-^  +    '   '    '    +  "n,      oio  5^  0. 

It  is  of  interest  in  this  connection  to  point  out  some  of  the  appU- 
cations  *  that  may  be  made  in  theoretical  physics  of  the  transforma- 
tion W  =  2". 

For  the  special  case  where  n  =  —  1,  we  have 

1 
w  =  -} 
z 

or 

j_  •  1 

x  +  iy 
whence 

X  —  y  /-v 

^  =  ^4:72'      '  =  ^^:ff'  (i> 

Corresponding  to  the  lines  w  =  c  we  have 

X        _ 

or 

c(a;^  -\-  y^)  —  X  =  0. 

This  equation  is  represented  by  a  system  of  circles  having  their  cen- 
ters on  the  X-axis  and  all  passing  through  the  origin.  For  the  or- 
thogonal system,  we  have 

or 

c(x^  +  2/^)  +  y  =  0, 

*  See  Jeans,  Electricity  and  Magnetism,  p.  261  at  seq;  Lamb,  Hydrodynamics, 
S''  Ed.,  p.  66. 


118 


MAPPING,  ELEMENTARY  FUNCTIONS 


[Chap.  IV. 


which  is  represented  by  a  system  of  circles  having  their  centers  on 
the  y-axis.     These  two  systems  of  circles  are  shown  in  Fig.  49. 

When  a  fluid  flows  over  a  plane  surface,  any  point  P  from  which 
the  fluid  flows  out  in  all  directions  in  a  uniform  manner  is  called  a 
source.  By  the  strength  of  the  source  is  understood  the  total  flow 
in  a  unit  of  time  across  a  small  closed  curve  about  the  source,  that  is, 


Fig.  49. 


the  time  rate  of  the  supply  of  the  fluid  through  the  source.  A 
negative  source  is  called  a  sink.  Let  P'  be  a  sink  and  suppose  it 
to  be  of  equal  strength  with  the  source  P.  Denote  this  common 
strength  by  m.  If  we  now  think  of  P  as  approaching  P'  in  such  a 
manner  that  the  product  m  •  PP'  remains  constant,  say  equal  to  n, 
then  we  say  that  we  have  a  plane  doublet  *  of  strength  n.  The 
velocity-potential  due  to  the  doublet  is  given  by 


</'(a5,  y)  = 


■nx 


x^  -f  y^ 


\ 


•  See  Pierce,  Newtonian  Potential  Function,  p.  434,  et  seq. 


/ 


^ 


Art.  28.]  THE   FUNCTION  2"  119 

while  the  lines  of  flow  are  given  by 

^fe  V)  =  Ji^,  =  c. 

By  comparing  these  functions  with  the  conjugate  functions  u(x,  y), 
v{x,  y)  given  in  the  equations  (1),  it  will  be  seen  that  u,  v  determine 
the  velocity-potential  and  the  lines  of  flow  respectively  of  a  plane 
doublet  at  the  origin  whose  strength  n  is  —1.  The  X-axis  is  the 
axis  of  the  doublet.  The  lines  of  flow  v  =  c  are,  as  we  have  seen, 
the  system  of  coaxial  circles  having  their  centers  on  the  F-axis, 
while  the  system  of  coaxial  circles  having  their  centers  on  the  X-axis 
are  the  lines  of  equal  velocity-potential. 

Writing  the  given  function  ty  =  2"  in  the  form 

u-\-  iv  =  p"(cos  0  -f- 1  sin  dY; 

we  have,  upon  equating  the  real  parts  and  likewise  the  imaginary 
parts, 

w  =  p"  cos  nd,        V  =  p^  sin  nB. 

For  n  =  1  the  first  of  these  equations  gives,  for  the  flow  of  an  in- 
compressible fluid,  a  system  of  equipotential  curves  parallel  to  the 
Y-axis,  and  the  second  gives  as  the  corresponding  lines  of  flow  a 
system  of  Unes  parallel  to  the  X-axis. 

It  has  been  pointed  out  that  when  n  =  2,  then  the  equation  u  =  c 
gives  a  system  of  rectangular  hyperbolas  having  the  axes  of  coordi- 
nates as  their  principal  axes.  In  the  applications  to  the  flow  of  an 
incompressible  fluid,  these  curves  are  the  lines  of  equal  velocity- 
potential  of  an  irrotational  fluid  having  constant  density  and  a 
steady  flow.  The  curves  v  =  c,  that  is  the  lines  of  flow  are  likewise 
a  system  of  rectangular  hyperbolas,  having  in  this  case  the  axes  of 

coordinates  as  asymptotes.     The  lines  d  =  0,  6  =  ^  are  parts  of  the 

same  line  of  flow  corresponding  to  y  =  0;  hence,  we  may  take  the 
positive  parts  of  the  coordinates  axes  as  fixed  boundaries,  and  thus 
obtain  a  case  of  irrotational  fluid  motion  in  an  angle  between  two 
perpendicular  walls  (see  Fig  37). 

By  the  proper  selection  of  the  value  of  n,  we  may  so  change  the 
conditions  as  to  represent  an  irrotational  niotion  of  the  fluid  be- 
tween two  rigid  walls  making  a  given  angle  0  with  each  other.  The 
lines  of  flow  are  given  by  the  equation 

p"  sin  72^  =  c, 


7 


120 


MAPPING,  ELEMENTARY  FUNCTIONS 


(Chap.  IV. 


where  the  Unes  6  =  0,  6  =  -  are  parts  of  the  same  line  of  flow,  namely 
the  one  by  putting  c  =0.     If  now  we  so  select  n  that  the  angle  - 

shall  be  the  required  angle  0,  that  is  if  n  =  - ,  we  get  as  the  required 

equations  of  the  equipotential  curves  and  lines  of  flow,  respectively, 
the  following: 


^ZaW 


Ex.  Find  the  value  of  n  such  that  the  function  w  =  z^  shall  determine  an 
irrotational  motion  of  a  liquid  between  two  walls  making  an  angle  of  60°  with 
each  other.  Find  the  velocity-potential  and  direction  of  the  flow  at  the  point 
«o  =  2  (cos  20°  +  t  sin  20°),  assuming  the  flow  to  be  steady.  Trace  the  equi- 
potential curve  through  the  given  point. 

We  have 

n  =  -  =  3, 


and  therefore  w  =  ^  =  u-\-w, 

whence  «  =  a;*  —  3  xy^,        v  =  3x^  —  y*. 

The  velocity-potential  at  the  given  point  is 

u  =  p*  cos  3  0 
=  2»co83-20°  =  4. 

The  equipotential  curve  is  then  given  by  the  equation 

a:»  —  3  zy*  =  4.  i;  tc 


Art.  28.]  THE  FUNCTION  2"  121 


From  the  definition  of  velocity-potential,  we  know  that  the  components  of  the 
velocity  in  the  direction  o 
The  velocity  v  is  given  by 


velocity  in  the  direction  of  the  X-axis  and  F-axis  are  —  v~>  ~"  ^>  respectively 


-^     V  =  - t—--  (8) 

yy  dx      dy 

We  can  determine  graphically  the  velocity  by  means  of  the  derivative  DzW.  As 
we  have  seen,  Art.  21, 

By  comparison  of  (8)  and  (9),  it  will  be  seen  that  the  point  P'  representing  the 
velocity  at  2o  is  the  reflection  upon  the  axis  of  imaginaries  of  the  point  P  repre- 
senting DtW.    We  have 

D^tp  =  3  zo"  =  12  (cos  40°  -I-  i  sin  40°). 

The  point  P  is  foimd  by  laying  off  on  a  line  making  an  angle  of  40°  with  the  axis 
of  reals  the  distance  OP  =  12.  By  reflection  upon  the  F-axis  the  point  P'  is 
found,  which  represents  the  velocity  at  Zo.  Drawing  from  Zo  a  line  parallel  to 
OP',  we  have  the  direction  of  the  flow  at  Zq.  This  flow  is  in  the  direction  of  the 
normal  to  the  equipotential  curve  through  Zo  as  we  should  expect. 

It  should  also  be  observed  that  if  the  point  Zo  is  allowed  to  move  along  a 
ciuve  of  flow 

V  =  3  xhf  —  y^  =  k, 

the  point  P'  varies  in  such  a  manner  as  to  describe  the  hodograph*  of  the  motion 
of  Zo,  and  thus  the  velocity  of  P'  determines  the  magnitude  and  direction  of  the 
acceleration  of  Zo- 

The  relation  between  the  velocity  in  the  Z-plane  and  the  deriva- 
tive DzW,  as  brought  out  in  the  above  exercise,  is  perfectly  general 
and  may  be  applied  to  any  case,  so  long  as  the  function  w  =  f(z) 
is  holomorphic  in  the  region  under  consideration.  It  should  also  be 
noted  in  this  connection  that  the  velocity  of  a  point  moving  along 
the  line  t>  =  c  in  the  TT-plane  is  always  the  negative  of  the  square  of 
the  speed  of  the  corresponding  point  moving  along  the  curve  of  flow 
in  the  Z-plane;  for,  denoting  by  Vw,  vz,  the  velocities  in  the  TF-plane 
and  Z-plane  respectively,  we  have  by  equation  (13),  Art.  27, 


Vw 


=  DzW 


_  /du        du\  /     du  _  .  du\ 
\dx)       \dy) 


|2  =    _  r2 


=  —  r^ 

where  r  is  the  speed  along  the  curve  of  flow  in  the  Z-plane. 
*  See  Ziwet,  Theoretical  Mechanics,  Part  I,  p.  80. 


122  MAPPING,  ELEMENTARY  FUNCTIONS  [Chap.  IV. 

29.  Definition  and  properties  of  e*.  We  shall  now  define  the 
exponential  function  e',  where  ^  is  a  complex  variable.  The  defini- 
tion and  properties  of  the  function  e',  where  x  is  a  real  variable, 
can  not  be  assumed  to  hold  when  the  variable  is  complex.  The 
function  e'  should  be  so  defined,  however,  as  to  include,  as  a  special 
case,  the  function  e'.  It  will  assist  us  in  formulating  this  definition 
to  recall  the  definition  and  some  of  the  general  properties  of  e*. 
The  number  e  is  defined  by  the  limit 


L   f  1  +  -V  =  2.7182818  .  .  .  , 
n=«  V        nj 


where  n  takes  all  positive  real  values.    Likewise,  the  exponential 

function  e'  is  defined  as  the  limit  L  [1  -\ — )  . 

n=oo  V        n} 

This  function  obeys  the  general  law, 

/(Xl)-/(X2)=M  +  X2).  (1) 

Differentiating  the  function  f{x)  =  e',  we  have 

D.fix)  =^  =  6-.  (2) 

We  shall  define  the  exponential  function  of  a  complex  variable 
by  the  relation, 

where  w  is  a  positive  integer.  We  shall  show  that  this  limit  exists 
for  all  finite  values  of  z  and  that  the  function  thus  defined  has  the 
general  properties  expressed  for  e^  in  (1)  and  (2), 

To  show  the  existence  of  this  limit,  we  proceed  as  follows.    We 
write 

i  +  i^i+^+iy 


Putting 
we  get 


n  n 

=  l  +  -  +  t^-  (4) 

n        n 


1  +  -  =  pcosd,       -  =  psind,  (5) 

n  n 

(l  +  -  j   =  [p(cos  d-\-isme)Y 

=  p"(cos  TiB  +  iam  nd).  (6) 


Art.  29.] 


THE  FUNCTION  e' 


123 


Since  n  can  be  taken  so  large  that  1  +  - ,  and  therefore  cos  6,  is 

n 

always  positive,  from  (5)  we  obtain  d  as  the  principal  value  of 


arc  tan  — 7 — ,  and 
n-\-  X 


whence 


The  limit  in  (3)  may  then  be  written 


(7) 


^     1  +  J   =^ 

n=oo   \  It'/  n=oo 


( cos  n  arc  tan  — ~ h  *  sin  n  arc  tan  — r — )  i 

\  n-^x  n-\-x/) 


n  -\-  X 

y2         \2 


=  L  I1  +  -]   '  L  (1  +  7 — 7 — r^J    •     L  cos  n  arc  tan —y— 
„=oo  \        n/     „=oo  V        (n  +  x)7      Ln='»  n  +  x 


+  1  L  sin  n  arc  tan  - 

n=oo  W  ~p  X 


provided  each  of  these  limits  exist.  These  limits,  however,  do  ex- 
ist and  can  be  readily  evaluated.  We  have  from  functions  of  a 
real  variable 


L     1  + 


-     =  e*. 
nj 

For  y  =  0,  the  second  limit  in  (8)  is  one;  for  y  5^  0,  we  have 
However,  we  have 


(9) 


(n+i)«>| 


L 

n=  00 


(n  +  rc)2 


r 

II 
e 


n=oo       (n+z)2 
II 

0 


=  e". 


124  MAPPING,  ELEMENTARY  FUNCTIONS  [Chap.  IV. 

Hence,  from  (9)  we  get 


i.h(;r^J=(^')'--  ''■ 


Since  the  cosine  is  a  continuous  function,  we  may  write  the  third 
limit  in  (8)  in  the  form 

L  cos  n  arc  tan  — r —  =  cos  L  n  arc  tan  — 7 —  o 

R.oo  n  +  x  n^-oo  n-j-x  v^ 

2/      /  ^ 
arc  tan  — 7 —  / 

=  cos  L  — f- ^     =  cosy.      (10) 

„=„  n  -\-  X  y 

n-\-  X 
Similarly,  we  have  for  the  final  limit  in  (8) 

L  sin  n  arc  tan —-^— =  sin  y.  (11) 

Hence,  substituting  these  results  in  (8)  we  obtain 

c*  =  c*(cos  1/  +  i  sin  y).  (12) 

This  result  not  only  shows  that  the  limit  given  in  (3)  exists  for  all 
finite  values  of  z,  but  it  gives  a  very  convenient  form  of  the  defini- 
tion of  the  exponential  function  e'. 

From  this  form  of  the  definition,  we  can  deduce  a  convenient 
method  for  writing  the  complex  number 

z  =  p(cos  d  -\-i  sin  d) ;  (13) 

for,  putting  x  =  0  in  (12),  we  have 

e*"  =  cos  y  -\-  i  sin  y, 
or  writing  this  result  in  the  usual  form,  we  have 

e«»  =  cos  6  -\-  i  sin  6. 

Hence  we  may  write  (13)  in  the  form 

z  =  pe'fl,  .      (14) 

&  form  of  expression  that  is  often  convenient. 

The  function  e*  is  uniquely  determined;  for,  we  have  from  (12) 

e'  =  u-\-  iv  =  e^  cos  y  -\-  ie'sin.  y, 
whence 

u  =  e'  cos  y,        v  =  ^siny.  (1 


Akt.  29.]  THE   FUNCTION   6'  125 

From  these  equations  it  follows  that  the  conditions  that  f{z)  =  e'  is 

an  analytic  function  are  satisfied;  for,  we  have  for  all  finite  values 

of  X,  y 

du  _  dv  du dv 

dx       dy'         dy  dx 

Therefore,  e'  is  holomorphic  in  the  finite  region  of  the  complex  plane 
and  consequently  is  an  analytic  function.  Hence,  in  the  finite  region 
e'  is  continuous  and  has  a  continuous  derivative.  Moreover,  e'  is 
a  single- valued  function  of  z,  and  e'  appears  as  a  special  case. 

From  (12)  it  can  be  shown  that  the  general  properties  of  the 
exponential  function  of  a  real  variable  may  be  extended  to  the  case 
where  the  variable  is  complex.  For  example  we  may  deduce  as 
follows  the  general  law  expressed  in  (1),  namely, 

Substituting  the  values  of  e^',  e^',  as  defined  by  (12),  we  have 

e^i .  e^i  =  gii  (cos  yi  +  i  sin  yi)  e^»(cos  ?/2  +  i  sin  1/2) 
=  e»>+^»  lcos(i/i  +  yz)  +  i  sin  (yi  +  1/2)  I 

The  law  of  differentiation  stated  in  (2)  holds  also  where  the  vari- 
able is  complex.     Remembering  that 

T^  du  ,    .dv 

ax        ax 
we  have  from  (12) 

DzB'  =■  e^  cos  y  -\-  ie"  sin  y 
=  e^(cos  ?/  +  I  sin  y) 
=  e^ 

It  is  to  be  observed  that  e'  is  a  periodic  function;  that  is,  the 
function  remains  invariant  when  z  is  replaced  by  z  plus  some  con- 
stant, say  CO.     Such  a  function  satisfies  the  relation 

/(2  +  CO)    =/(.). 

The  constant  co  is  called  a  period  of  the  given  function.  A  periodic 
function  takes  all  of  its  values  as  the  variable  z  takes  the  values  in  a 
definite  region  of  the  complex  plane,  known  as  the  region  of  peri- 
odicity, and  repeats  those  values  as  z  varies  over  another  equal 
portion  of  the  plane.  In  this  particular  case  the  regions  of  peri- 
odicity are  parallel  strips  bounded  by  lines  parallel  to  the  axis  of 


126 


MAPPING,  ELEMENTARY  FUNCTIONS 


[Chap.  IV. 


reals  and  at  a  distance  of  2  t  from  each  other.  We  say  then  that 
the  function  has  the  period  2  in.  To  show  this  to  be  the  case,  we 
may  write 

=  e*  \cos{y  +  2t)  +tsm(2/  +  2t)} 
=  e^  {cosy  +  isiayl 
=  e. 

Instead  of  2  ti,  we  could  have  taken  equally  well  any  multiple  of  it 
as  a  period.  It  would  not  have  answered  the  purpose  to  have 
taken  a  fractional  part  of  2  7rt  nor  any  number  less  than  2  vi;  that 
is  2  7ri  is  the  smallest  constant  that  answers  the  purpose.  We  in- 
dicate this  fact  by  calling  2  id  the  primitive  period  of  the  function. 
When,  as  in  this  case,  all  of  the  periods  are  multiples  of  a  single 
primitive  period,  the  function  is  called  a  simply  periodic  function. 
Let  us  now  undertake  to  map  the  Z-plane  upon  the  W-plane, 
and  conversely,  by  means  of  the  relation  w  —  e^  Since  the  given 
function  is  holomorphic  having  a  derivative  different  from  zero  for 
finite  values  of  z,  it  follows  that  the  mapping  is  conformal  in  the 


Fig.  51. 


Fig.  52. 


finite  region;  that  is,  in  this  region  the  similarity  of  infinitesimal 
elements  is  preserved.  We  shall  make  use  of  the  fact  that  the 
function  is  periodic,  having  the  period  2  id.  If  we  draw  through 
the  points  id  and  —  tti  in  the  Z-plane  two  lines  parallel  to  the  X-axis, 
the  region  bounded  by  these  hues  maps  into  the  entire  IF-plane 
and    therefore   may   be    taken   as    the   fundamental   region.     In 


Art.  29.]  THE  FUNCTION  6'  127 

mapping  we  shall  consider  the  boundary  line  y  =  t,  but  not  the 
boundary  line  y  =  —  t,  &s  belonging  to  the  fundamental  region. 
What  is  said  of  this  region  may  be  said  of  any  one  of  the  regions 
bounded  by  the  lines 

y  =  {2k-j-l)T,    y  =  {2k-l)T,    A:  =   •  •  •  ,  -2,  -  1,  0,  1,  2,  •  •  •  . 

The  line  w  =  0  is  the  map  of  certain  lines  parallel  to  the  X-axis. 
To  show  this,  put  w  =  0  in  (15)  and  thus  obtain 

0  =  6''  cos  y,  (16) 

whence  for  finite  valuesof  x,  we  have  w  =  0  as  the  map  of  the  lines 

y  =  (2fc  +  l)|,     A:  =  -  •  .  ,  -2,  -1,0,  +1,  +2,  .  .  .  .   (17) 

Within  the  fundamental  region  —  ir  <  y  =  r,  we  have  the  lines  cor- 
responding to  A;  =  0  and  A;  =  —  1 ;  and  hence  the  line  m  =  0,  that  is  the 
F-axis,  is  the  map  of  two  lines  of  the  Z-plane  lying  within  this  region, 
namely: 

y  ='^     and    y=-\-  (18) 

For  i/=  X  we  have  from  (15)  v  =  e',  and  for  ?/  =  —  -  we  have  v  =—  e'; 

IT 

so  that  the  positive  F-axis  is  the  map  of  the  line  y  =  ^,  while  the  nega- 

•  TT  TT 

tive  F-axis  is  the  map  of  the  line  y  =  —  ^ .  Since  for  a:  >  0,  ?/  =  db  ^  we 

have  I  e^  I  =  e^  >  1,  it  follows  that  the  portion  of  the  positive  F-axis 
exterior  to  the  unit  circle  about  the  origin  is  the  map  of  the  positive 

TT  • 

half  of  the  line  y  =  k,  and  the  portion  of  the  negative  F-axis  exte- 
rior to  the  unit  circle  is  the  map  of  the  positive  half  of  the  line 
y  =  —  - .     Likewise,  since  for  a;  <  0,  ?/=  db  7;  we  have  \  e'  \  =  e'  <  1, 

it  follows  that  the  negative  halves  of  the  lines  y  =  5 ,  y  =  —  ^c  map 

respectively  into  the  portions  of  the  positive  and  the  negative  F-axis 
lying  within  the  unit  circle. 

In  a  similar  manner  we  have  from  (15),  for  f  =  0, 

0  =  e^  sin  y,  (19) 


128 


MAPPING,  ELEMENTARY  FUNCTIONS 


[Chap.  IV. 


and  hence  the  line  y  =  0,  that  is  the  ?7-axis,  is  the  map  of  the  lines 
y  =  kir,    fc  =  .  .  •  ,  -2,  -  1,  0,  +  1,  +  2,  •  •  •  .  (20) 

For  the  fundamental  region  —  t  <  y  =  t,  we  have  A;  =  0,  1,  and 
consequently 


y  =  0,Tr. 
For  these  values  of  y,  we  obtain 

u  =  e',    u  =  —  e', 


(21) 


(22) 


respectively.  Hence  the  positive  tZ-axis  is  the  map  of  the  X-axis, 
and  the  negative  C/-axis  is  the  map  of  the  line  y  =  tt. 

The  positive  halves  of  the  lines  y  =  0  and  y  =  t  map  into  that 
portion  of  the  U-axis  exterior  to  the  unit  circle,  while  the  negative 
halves  of  the  same  lines  map  into  that  portion  of  the  C/-axis  within 
the  unit  circle. 

Any  line  parallel  to  the  X-axis  maps  into  a  half -ray  in  the  TF-plane 
proceeding  from  the  origin,  Fig.  53.     This  may  be  shown  as  follows. 


Fig.  53. 


Fig.  54. 


Eliminating  x  from  equations  (15)  by  multiplying  the  first  of  these 
equations  by  sin  y  and  the  second  by  cos  y  and  subtracting,  we  have 

usiay  —  V  cos  y  =  0. 

For  constant  values  of  y,  this  equation  gives  straight  lines  of  the 
form 

V  =  mu, 

where  m  =  tan  y.  Since  e'  is  positive  for  all  finite  values  of  x,  it 
follows  from  (15)  that  any  line  y  =  c  maps  into  a  half-ray  from  the 
origin  taken  along  the  line  v  =  mu;    the  portion  of  this  half -ray 


Art.  29.]  THE  FUNCTION  e'  129 

interior  to  the  unit  circle  corresponds  to  negative  values  of  x,  while 
the  portion  exterior  to  the  unit  circle  corresponds  to  positive  values 
of  X.  If  successive  values  of  y  differ  by  equal  amounts,  then  the 
corresponding  half-rays  in  the  TT-plane  will  make  equal  angles  with 
each  other. 

The  map  of  a  line  parallel  to  the  F-axis  may  be  easily  obtained  as 
follows.     Eliminating  y  from  the  equations 

u  =  e^  cos  y,     v  =  e^  sin  y 
by  squaring  and  adding,  we  have 

^2    _|_    j;2    _    g2x    ^gQg2  y    _|_    sjjj2  y) 

For  any  constant  value  of  x,  we  have  then  a  circle  in  the  W-plane 
about  the  origin  as  a  center. 

For  x  =  0, 

we  have  w^  +  y^  =  1 ; 

that  is,  the  F-axis  maps  into  the  unit  circle  about  the  origin  in  the 
TT-plane.  For  a:  =  c  >  0,  the  map  in  the  W-plane  is  a  circle  exte- 
rior to  the  unit  circle;  and  for  x  =  c  <  0,  the  map  is  a  circle  lying 
within  the  unit  circle.  From  what  has  been  said,  it  will  now  be  seen 
that  the  regions  a,  6,  c,  d.  A,  B,  C,  D,  Fig.  52,  map  respectively  into 
the  regions  a,  b,  c,  d,  A,  B,  C,  D,  Fig.  51,  the  lower  bank  of  the  line 
y  =  IT,  and  the  upper  bank  of  the  line  y  =  —  t  mapping  respectively 
into  the  upper  and  the  lower  banks  of  the  negative  t/-axis. 
Any  line  y  =  mx 

passing  through  the  origin,  other  than  the  axes  of  coordinates,  maps 
into  a  curve  in  the  T^-plane  that  cuts  the  half-rays  from  the  origin 
at  a  constant  angle;  that  is,  it  maps  into  a  logarithmic  spiral  about 
the  origin.  Since  the  point  x  =  0,  y  =  0  maps  into  the  point 
M  =  1,  t)  =  0,  all  of  these  logarithmic  spirals  pass  through  the  point 
u  =  1,  V  =  0. 

As  we  have  seen  the  whole  of  the  TT-plane  can  be  mapped  into  any 
one  of  a  number  of  strips  parallel  to  the  axis  of  reals  in  the  Z-plane. 
Suppose  we  map  it  into  the  fundamental  region  lying  between  the  lines 
y  =  IT,  y  =  —IT.  Consider  the  line  w  =  c,  c  >  0.  For  this  value  of 
u,  we  have 

c  =  e'  cos  y, 

or  y  =  arc  sec  —  •  (23) 


130 


MAPPING,  ELEMENTARY  FUNCTIONS 


[Chap.  IV. 


If  we  put  e*  =  c,  that  is  x  =  log  c,  we  have 

y  =  arc  sec  1  =  0. 

The  curve  whose  equation  is  (23)  then  cuts  the  X-axis  at  the  point 
X  =  log  c.     As  X  increases  y  increases  and  approaches  asymptoti- 

cally  the  line  2/  =  « •     The  sign  of  y  is  determined  by  the  sign  of  v 


2' 


in  the  relation 


t>  =  e^  sin  y. 


(24) 


Smce  e'  is  always  positive,  sin  y  and  therefore  y  is  positive  or  nega- 
tive according  as  v  is  positive  or  negative.     The  curve  is  therefore 


,«<K<sw.<:4jJ 


*^'vV«NSSm^V.V.W+\W 


,  !  i 


Fig.  55. 


i^ 


U 


Fig.  56. 


symmetrical  with  respect  to  the  X-axis  and  is  situated  as  is  indicated 
in  Fig.  56.  As  c  is  assigned  dififerent  values,  the  point  where  the 
curve  crosses  the  X-axis  changes.  For  c  >  1,  the  curve  cuts  the 
X-axis  to  the  right  of  the  point  x  —  0;  for  c  =  1,  it  crosses  at  the 
origin;  for  0  <  c  <  1  it  crosses  to  the  left  of  the  origin. 

It  will  be  remembered  that  the  negative  half  of  the  U-axls  maps 
into  the  lines  y  =  t,  y  =  —  x.  That  portion  of  the  line  u  =  c,  where 
c  <  0,  lying  above  the  [/-axis  will,  as  we  have  seen,  map  into  the 

TV 

portion  of  the  fundamental  region  lying  between  the  lines  y  =  «  ^^^ 

y  =  T.  Moreover,  since  c  is  now  negative  it  follows  that  y  decreases 
as  X  increases.     The  form  of  the  curve  is  indicated  in  Fig.  56.     It  is 

asymptotic  to  the  line  2/  =  5 .     In  the  same  way  it  follows  that  the 

portion  of  m  =  c  lying  below  the  C/-axis  maps  into  a  curve  begin- 
ning on  the  line  y  =  —k  and  becoming  asymptotic  to  the  straight 

TC        • 

line  y  =  —  rt  >  since  the  ordinate  increases  with  x.   The  results  of  map- 


Art.  29.]  THE   FUNCTION   e'  131 

ping  the  TT-plane  upon  the  fundamental  region  —ir<y  =  x 
are  of  course  repeated  in  any  other  strip  bounded  by  the  lines 
2/  =  (2A;  +  l)7r,  y  =  (2fc-l)7r. 

It  is  of  interest  in  this  connection  to  observe  the  form  of  the 
surface  f  =  u(x,  y).  The  curves  just  obtained  by  mapping  upon 
the  Z-plane  the  lines  w  =  c  are  the  curves  of  intersection  of  this 
surface  by  the  plane  ^  =  c.  A  general  notion  of  the  form  of  the 
surface  is  obtained  by  noting  the  manner  in  which  u  changes  as  x 
increases  along  certain  lines  parallel  to  the  X-axis.  Take  for  this 
purpose  the  lines 

IT  IT 

y   =    -TT,        -■^,        0,        -,       TT. 

From  (15)  it  will  be  seen  that  along  these  lines,  we  have 
w  =  -  e',     0,     e',     0,     -e'. 

As  X  decreases  without  limit  through  negative  values,  each  of  these 
values  of  u  approaches  zero.     However,  as  x  increases  the  value  of 

u  remains  zero  along  the  lines  y  =  —  ^,   5,  but  increases  without 

limit  along  the  line  y  =  0,  and  decreases  without  limit  along  the 
lines  y  =  —  tt,  tt.  Hence,  we  have  a  surface  that  is  flat  at  the  extreme 
left  and  towards  the  right  has  ridges  and  valleys  of  increasing  magni- 
tude. These  ridges  and  valleys  extend  parallel  to  the  axis  of  reals 
and  their  magnitude  is  readily  determined  by  taking  a  cross-section 
of  the  surface  parallel  to  the  F-axis. 

As  an  illustration,  let  us  consider  the  function 

z  =  w  -\-  e^. 

As  we  shall  see  later,  this  function  is  of  importance  in  the  consideration  of  certain 
problems  in  mathematical  physics.  We  shall  map  a  given  configuration  from  the 
TT-plane  upon  the  Z-plane  by  means  of  this  relation.  In  order  to  do  so,  we  must 
first  obtain  x  and  y  in  terms  of  u  and  v.    Writing  the  given  function  in  the  form 

X  -\-iy  =  u  +iv  -\-  €"+»" 
=  M  -f-  IV  +  e"  •  e'" 
=  M  +  if  +  e"  (cos  V  -\-%  sin  v), 

we  have  upon  equating  the  real  and  the  imaginary  parts 

X  =  ti  +  e"  cos  V,         y  =  V  -\-  e"  sin  v.  (25) 

The  axis  v  =  0  maps  into  the  X-axis;  for,  we  have  in  this  case  from  the  equa- 
tions (25) 

X  =  M  +  e",     2/  =  0, 


132 


MAPPING,   ELEMENTARY  FUNCTIONS  [Chap.  IV. 


and,  consequently,  every  point  on  the  C7-axis  maps  into  a  point  on  the  X-axis, 
the  point  «;  =  0  mapping  into  z  =  +1  (Fig.  57).     For  «  =  t,  we  have 


X  =  u  —  e" 


y  =ir. 


As  w  moves  along  the  line  v  =  v  from  u  =  —  oo  to  u  =  0,  the  corresponding  point 
in  the  Z-plane  moves  along  the  line  y  =  v  from  x  =  —  oo  to  x  =  —  1.  As  the 
value  of  t*  continues  to  increase  from  u  =  0  to  u  =  qo  ,  the  value  of  x  passes  from 


U 


Fig.  57. 


Fig.  68. 


X  =  —1  back  to  X  =  — 00 ,     We  say  that  the  line  v  =  ir  maps  into  the  line  y  =  x 
in  such  a  manner  that  the  line  doubles  back  upon  itself  at  the  point  x  =  —  1. 

In  a  similar  maimer,  the  line  v  =  —n-  maps  into  the  line  y  =  —v,  bending 
back  upon  itself  at  the  point  x  =  —  1. 


Let  us  now  consider  the  line  v  = 


For  this  value  of  v  we  have 


X  =u,     y  =  2  +  «"• 
For  M  =  —  00 ,  we  have  x  =  — oo,y  =  ^.     As«  increases,  y  increases  until  u 

reaches  the  value  zero,  where  x  =  0,  2/  =  |  +  l.    Asu  increases  through  positive 
values,  y  continues  to  increase  with  u  as  indicated  in  the  figure. 
For  values  of  v  lying  between  ^  and  tt,  say  for  (t  -«),  we  have 

X  =  u  —  e"  cos  «, 
J/  =  IT  —  «  +  e"  sin  6. 

From  these  equations,  we  have  for  u  =  —  oo, 

X  =   —00, 

1/  =  X  -  «. 


Art.  30.] 


THE  FUNCTION  LOG  Z 


133 


As  u  increases,  both  x  and  y  increase,  although  y  increases  very  slowly,  until  we 
have 

DuZ  =  1  —  e"cos«  =  0; 
that  is,  until 

e"  CO&  t  —  1. 

As  e"  cos  c  becomes  greater  than  1,  DuX  becomes  negative  and  x  decreases  while  y 
continues  to  increase  and  that  more  rapidly.  The  general  form  of  such  a  curve 
is  shown  in  the  figure. 

For  values  of  c  lying  between  0  and  - ,  the  line  v  =  c  maps  into  a  curve  such 

that  y  at  first  increases  very  slowly  and  then  more  rapidly  as  u  takes  on  large 
positive  values.     In  this  case,  however,  x  also  continues  to  increase  as  u  increases. 

For  values  of  v  less  than  0,  the  curves  lie  below  the  X-axis  and  are  symmetrical 
as  to  that  axis  with  those  already  obtained.  The  mapping  of  these  curves  there- 
fore presents  nothing  new. 

The  given  function  expresses  the  motion  of  a  fluid  from  a  reservoir  of  indefi-    , 
nitely  large  size  into  a  narrow,  restricted  channel  bounded  by  thin  parallel  walls.*-^' 
If  the  sign  of  w  is  changed,  the  given  function  represents  the  flow  as  taking  place    ' 
in  the  opposite  direction.     As  may  be  shown,  the  velocity  of  the  flow  increases 
indefinitely  in  the  neighborhood  of  the  points  (  —  1,  x),  (  —  1,  — ir).  '" 

30.  The  function  w  =  log  z.  We  shall  now  define  the  logarith- 
mic function  and  discuss  some  of  its  properties.  In  real  variables 
the  logarithm  is  frequently  defined  as  the  inverse  function  of  the 
exponential.  This  property  will  be  used  in  defining  the  logarithm 
of  a  complex  number.  The  general  properties  of  inverse  functions 
have  been  discussed  in  another  connection.  To  determine  the  in- 
verse of  the  exponential  function  let  us  consider  the  relation 

e"  =  z.  (1) 

We  have,  as  elsewhere, 

w  =  u-\-  iv, 
z  =  p(cos  0  -f-  i  sin  ^). 


Equation  (1)  may  then  be  written  in  the  form 


(2) 


Remembering  that 

e'"  =  cos  V  -\-i  sin  y, 

we  may  now  write  (2)  in  the  form 

e"(cos  t;  +  i  sin  v)  =  p(cos  <^  -f  i  sin  <^). 

Equating  the  real  and  the  imaginary  parts  in  this  equation,  we 
obtain 

e"  cos  V  =  p  cos  <^,     e"  sin  v  =  p  sin  </>.  (3) 

*  See  Lamb,  Hydrodynamics,  3''  Ed.,  p.  70. 


134  MAPPING,   ELEMENTARY  FUNCTIONS  [Chap.  IV. 

Each  number  entering  into  these  equations  is  real,  and  consequently 
we  can  solve  the  equations  for  u  and  v  by  the  means  already  at  our 
disposal.  Squaring  each  member  of  these  equations  and  adding,  we 
have 

(e'*)2  (cos^  V  +  sin^  v)  =  p\cos^  <f>  +  sin^  <^), 
whence 

(e»)2  =  p2, 

but  as  e"  and  p  are  always  positive  numbers,  we  may  write 

e"  =  p.  (4) 

Making  use  of  this  relation,  we  have  from  (3) 

cos  V  =  cos  <f>, 
sin  i;  =  sin  0, 
whence 

V  =  </>.  (5) 

Since  u  and  p  are  both  real  numbers,  we  have  from  (4) 

u  =  logp  =  log  I  z  |. 

It  is  to  be  noticed  that  for  z  =  0,  and  therefore  p  =  0,  the  equation 
e"  =  p  has  no  finite  solution.  For  all  other  values  of  z,  the  equa- 
tions 

M  =  logp  =  log  I  2  I, 

V  =  <f>  =  amp  z 

determine  definite  values  of  the  coordinates  u,  v.  The  correspond- 
ing value  of  10  is  defined  as  the  logarithm  of  z.  We  have  then  as  the 
formal  definition  of  iw  =  log  2 

log  z  =  loff  p -\- i^,  (6) 

which  may  also  be  written  in  the  form 

log  z  =  log  I  2  I  +  t  amp  z.  (7) 

For  any  particular  point  of  the  complex  plane,  say  Zo,  there  are 
an  infinite  number  of  values  of  log  Zq  differing  from  each  other  by 
some  multiple  of  2iri.  This  result  is  a  consequence  of  the  peri- 
odicity of  the  exponential  function,  which  is  the  inverse  of  the  loga- 
rithmic function;  or,  it  follows  directly  from  the  definition  of  a 
logarithm,  for  since  we  have  the  same  point  z  if  ^  is  replaced  by 
(0  +  2  kw),  where  A;  =  1,  2,  .  .  .  ,  it  follows  from  the  definition  that 
log  z  has  an  infinite  number  of  values  for  this  same  value  of  z.    In 


Aht.  30.]  THE  FUNCTION  LOG  Z  135 

the  discussions  of  the  present  chapter,  unless  otherwise  stated,  0 
will  be  restricted  to  the  chief  amplitude  of  2,  and  for  such  a  value  of 
^,  log  p  +  i^  is  called  the  principal  value  of  the  logarithm.  In  a 
subsequent  chapter  we  shall  discuss  the  logarithm  as  a  multiple- 
valued  function,  thus  giving  to  <^  all  possible  values. 

The  logarithms  of  the  positive  real  numbers  appear  as  a  special 
case  of  those  of  complex  numbers,  because  for  such  numbers  the 
value  of  <^  is  zero.  The  logarithms  of  negative  numbers  may  now 
be  given  a  definite  significance;  for,  if  z  is  a  negative  real  number, 
we  have 

z  =  p(cos  TT  +  i  sin  tt), 
and  hence  we  obtain 

log  2  =  log  p  +  %-K, 

which  is  represented  by  a  definite  point  in  the  complex  plane. 

Except  for  2  =  0,  the  logarithmic  function  is  holomorphic  in  the 
finite  region;  for,  the  Cauchy-Riemann  differential  equations  are 
satisfied.     We  have 

w  =  u-\-iv  =  log  p  +  i(^ 

=  log  Vx^  +  2/^  +  i  arc  tan  -  • 

X 

Consequently,  we  obtain 


whence 


y 

du  X  dv 


u  =  log  Vx^  +  y^,         V  =  arc  tan  • 

X 


dx      0^  -\-  y^  dx  x^  +  y^ 

du  _      y  dv  _       X 

By      x'-  +  y"^  by      x^  +  y" 

Hence,  with  the  restriction  placed  upon  its  amplitude  the  function  is 
analytic.     This  conclusion  does  not  hold,  however,  for  x  =  0,  y  =  0, 

since  the  partial  derivatives  -r-  >  T"  >  t~  j  v   become  indeterminate 
^  dx    dy'  dx    dy 

in  this  case. 


In  putting 


V 
amp  2  =  ^  =  arc  tan  - , 


it  should  be  noted  that  care  must  be  taken  to  distinguish  between 

arc  tan  —  and  arc  tan .     We  may  not  replace  these  two  ex- 

a  —a 


136  MAPPING,  ELEMENTARY  FUNCTIONS  [Chap.  IV. 

pressions  by  arc  tan[  — ),  as  one  might  at  first  think  possible;  for, 

the  first  expression  is  the  amplitude  oi  z  =  a  —  ih,  while  the  second 
is  the  amplitude  of  z  =  —  a  +  t6.  These  two  values  of  z  have  the 
same  moduli,  but  their  ampUtudes  differ  by  r.     For  a  similar  reason, 

we  must  distinguish  between  arc  tan  — -  and  arc  tan  -  • 

The  function  log  z  obeys  the  laws  of  logarithms  for  real  variables. 
We  have,  for  example, 

log  Zi  +  log  22  ==  log  (ZiZi) ,  (8) 

where  Zx,  z^  are  different  from  zero. 

To  show  that  this  relation  holds,  we  have 

log  zi  +  log  22  =  flog  pi  +  ^0ll  +  Hog  p2  +  i<h\ 
=  flog  Pi  +  log  P2I  +  \i<l>i  +  i<h\ 
=  log  P1P2  +  i(0i  +  <h) 
=  log  (21Z2) ; 

since,  in  multiplying  two  complex  numbers  we  multiply  the  moduli 
and  add  the  ampUtudes. 

To  find  the  derivative  of  a  logarithm,  we  have 


and  therefore 


dw  _  du       .dv 
dz       dx        dx ' 


dlogz  _  d  ,  „,/  o   ,   „o   ,    .  d 


,       —  ^   log  y/x^  +  y^  +  1 7-  arc  tan  - 
dz  dx  dx  x 

_      X  ^y      _       1       _  1 

x^  +  y^      x''-  +  y^      X  -^-iy      z 


(9) 


The  mapping  of  the  Z-plane  upon  the  TF-plane  by  means  of  the 
logarithmic  function  gives  a  conformal  representation  because  the 
function  is  analytic.  The  only  finite  singular  point  is  the  origin. 
As  we  have  seen  the  logarithm  is  the  inverse  function  of  the  expo- 
nential function.  The  general  properties  of  the  configuration  ob- 
tained by  mapping  by  means  of  the  relation  w  =  log  2  follow  at 
once  from  the  earlier  discussion  of  the  mapping  by  means  of  the 
functional  relation  w  =  e^  The  only  difference  in  the  two  cases 
is  that  the  Z-plane  and  the  TF-plane  are  interchanged. 


Art.  30.]  THE  FUNCTION  LOG  Z  137 

Because  the  logarithmic  function  enables  us  to  map  a  system  of 
concentric  circles  and  their  orthogonal  rays  into  a  system  of  parallel 
lines  and  another  system  of  straight  lines  orthogonal  to  them,  it  is 
for  some  purposes  one  of  the  most  useful  of  the  functions  thus  far 
discussed.  One  of  the  important  applications  of  this  function  is  to 
that  system  of  map  drawing  known  as  Mercator's  projection.  If  we 
undertake  to  map  the  earth's  surface  upon  a  plane  by  means  of  a 
projection  from  the  north  pole  as  the  center  of  projection,  we  obtain 
a  corresponding  configuration  in  the  plane.  The  region  about  the 
north  pole  becomes  very  much  distorted  by  this  process.  It  is  often, 
desirable  in  navigation  so  to  direct  a  ship's  course  as  to  cut  the  suc- 
cessive meridians  at  a  given  angle,  that  is,  to  keep  the  ship  headed 
toward  a  definite  point  of  the  compass.  The  curves  indicating  the 
ship's  course  upon  the  earth's  surface,  known  as  loxodromes,  project 
into  logarithmic  spirals  upon  the  complex  plane,  when  that  plane  is. 
tangent  to  the  earth's  surface  at  one  of  the  poles  while  the  opposite 
pole  is  taken  as  the  center  of  projection.  By  means  of  the  logarith- 
mic function,  the  system  of  circles  and  orthogonal  rays  into  which 
the  earth's  surface  maps  by  this  method  of  projection  may  be  changed 
into  two  systems  of  parallel  straight  lines  orthogonal  to  each  other. 
In  this  new  system  the  meridians  become  a  system  of  parallel  straight 
lines,  while  the  parallels  of  latitude  become  a  system  of  parallels 
orthogonal  to  the  first.  All  loxodromic  curves  become  straight  lines 
cutting  these  two  systems  of  parallels  at  a  given  constant  angle. 
This  result  simplifies  the  problems 
of  navigation  by  the  use  of  the 
mariner's  compass. 

The  configuration  represented 
in  Fig.  59  arises  in  mathematical 
physics  whenever  we  have  a 
source  at  the  origin  and  a  sink  at 
an  infinite  distance.  The  rays 
are  in  this  case  the  lines  of  flow 

and  the  concentric  circles  are  lines  of  equal  velocity-potential.  If  we 
pass  through  the  origin  a  straight  wire  of  indefinite  length,  through 
which  a  current  of  electricity  is  passed,  we  have  in  any  plane  per- 
pendicular to  this  wire  an  induced  magnetic  field  likewise  repre- 
sented by  the  configuration  in  Fig.  59,  except  that  in  this  case  the 
pencU  of  rays  become  the  lines  of  equipotential  and  the  system  of 
concentric  circles  are  lines  of  force.     If  the  electric  current  flows 


138  MAPPING,  ELEMENTARY  FUNCTIONS  [Chap.  IV. 

through  the  wire  ui  the  direction  from  below  to  above  the  plane  of 
the  paper,  the  direction  of  the  magnetic  force  is  then  as  indicated  by 
the  arrow-heads.* 

z  —  1 
Consider  the  function  w  =  log  — r—r  .    Writing  the  given  function  in  the  form 

z  +  1 

u  +  w,  we  have  upon  separating  the  real  and  imaginary  parts 

U  =  log-^ 1— r4   =  log       .  '       '    "    ) 

V  =  amp  («  —  1)  —  amp  («  +  1)  ==  arc  tan  — ^-r  —  arc  tan     J_  ,  • 
For  ti  =  c,  we  have  

c  =  log     ,  t 

V(X  +  1)2  +  y2 

(X  -  1)2  +  y2 


e»«  = 


(X  +  1)2  +  2/2' 


x'  +  y'  +  2^^^x  +  l=0.  (10) 


This  equation  is  represented  by  a  system  of  coaxial  circles  having  their  centers 
3  of  these  circles  are  given  by  the  equati 


on  the  X-axis.    The  centers  of  these  circles  are  given  by  the  equations  LJ^'*''^ 


and  the  radii  are  equal  to 


^■r  ) 


v/(f^)'- 


or"  > 


For  negative  values  of  c  the  center  lies  to  the  right  of  the  origin.  The  point 
-(+1,  0)  lies  within  all  of  these  circles.  For  positive  values  of  c  the  centers  of  all 
the  circles  lie  to  the  left  of  the  origin  and  inclose  the  point  (  —  1,  0).  Correspond- 
ing to  c  =  0,  we  have  a  circle  of  infinite  radius,  that  is  a  straight  line  perpendicular 
to  the  segment  joining  the  points  (1,  0)  and  (  —  1,  0)  at  its  middle  point,  namely 
at  the  origin. 

For  V  =  c,  we  have 

y  V 

c  =  arc  tan  — ?-^  —  arc  tan 


X  —  \  a;  +  1 

whence 

y y_ 

.             x-\      x  +  1              2y 
tan  c  =  — ' 


,    ,       y"  a;2  +  J/2  -  1 

^'^x2_i 


^*  +  2/^-7^2/  =  l-  (11) 


tanc 
*  See  J.  J.  Thomson,  Electricity  and  Magnetism,  4*  Ed.,  p.  329 


Akt.  30. 


THE  FUNCTION  LOG  Z 


139 


This  is  the  equation  of  a  system  of  coaxial  circles  having  their  centers  on  the 
F-axis.  Each  of  these  circles  passes  through  the  points  (1,  0)  and  (  —  1,  0).  The 
general  form  of  the  configiiration  in  the  Z-pIane  is  given  in  Fig.  60. 

This  configuration  may  be  reproduced  in  the  physical  laboratory  as  follows. 
Given  a  glass  plate  covered  with  iron  filings.  Pass  long  straight  parallel  wires 
through  the  points  A  =  (+1,0)  and  B  =  (  —  1,  0)  perpendicular  to  the  plate 


>-X 


Fig.  60. 

and  allow  an  electric  current  of  equal  strength  to  flow  in  opposite  directions 
through  the  two  wires.  By  jarring  slightly,  the  filings  tend  to  arrange  them- 
selves along  the  system  of  circles  about  the  points  A,  B  and  having  their  centers 
upon  axis  of  reals.  These  circles  are  the  lines  of  magnetic  force.  The  direction 
of  this  force  depends  upon  the  direction  of  the  two  currents.  If  the  direction  of 
the  current  through  the  plane  of  the  paper  at  A  is  downward  and  that  through 
B  is  upward,  the  direction  of  the  force  is  as  indicated  in  the  figure.  The  orthog- 
onal system  of  circles  all  pass  through  the  points  A,  B  and  are  the  Unes  of  equi- 
potential. 

.  For  the  flow  of  incompressible  fluids,  we  obtain  the  same  configuration  when- 
ever one  of  the  points  A,  B  is  a.  source  and  the  other  a  sink,  both  being  of  the 
same  strength.  The  circles  through  A  and  B  are  then  the  lines  of  flow  and  the 
circles  of  the  orthogonal  system  are  the  lines  of  equal  velocity-potential.  If  A 
is  the  source  and  B  the  sink  then  the  direction  of  the  flow  is  from  A  to  B. 

For  the  function  w  =  log  (z  +  1)  (2  —  1),  we  have  a  different  configuration. 
From  the  given  fimction,  we  obtain 

u  +  iv  =  log\z  +  l\-\z  —  l\+i[  amp  (z+1)  +  amp  (z  —  1) } , 


V 


140 


MAPPING,  ELEMENTARY  FUNCTIONS 


[Chap.  IV. 


or  ti  =  log  1 2  +  1  I  •  I  z  -  1  I  =  log  V(x  +  1)»  +  j/»  V(x  -  1)2  +  y\ 

V  =  amp 
For  u  =  c,  we  get 


y  y 

V  =  amp  («  +  1)  +  amp  (z  —  1)  =  arc  tan      '  .  +  arc  tan 

X  -J-  1 


X  -  1 

x«  +  y«  +  2(x*  +  1)  y»  -  2x*  +  1  =  e^^ 
which  f or  c  7^  0  is  represented  by  a  system  of  Cassinian  ovals  as  shown  in  Fig.  61. 

n 


Fig.  61 


For  c  =  0  the  equation  represents  a  lemniscate  having  its  double  point  at  the 
origin.     For  the  orthogonal  system  of  curves,  we  have 


y      .  y 

c  =  arc  tan      .   ,  +  arc  tan 


whence 


x  +  l 


y    j^    y 


tanc  = 


x+l ' x-l 
1  - 


x-l' 
2xy 

X't  —  yi  -  1 


X*-  1 


X*  —  yi  — xy  =  1. 

'       tanc 


This  equation  gives  a  system  of  hjrperbolas  passing  through  the  two  points  (1,  0) 
and  (  —  1,  0)  as  indicated  in  Fig.  61. 

This  configuration  comes  into  consideration  in  theoretical  physics  whenever 
the  [>oints  A  and  B  are  sources  of  equal  strength.     If  we  are  considering  the  flow 


Aht.  30.]  .     THE   FUNCTION  LOG  Z  141 

of  an  incompressible  fluid,  the  curves  u  =  c  are  the  lines  of  equal  velocity-potential, 
and  the  curves  v  =  c  are  the  lines  of  flow.  If,  however,  we  are  considering  a 
magnetic  field,  induced  by  passing  currents  of  equal  strength  in  the  same  direc- 
tion through  parallel  straight  wires  intersecting  the  plane  at  A  and  B,  the  curves 
u  =  c  become  the  lines  of  force,  and  v  =  c  are  the  lines  of  equipotential.  Ordi- 
narily a  line  of  force  does  not  intersect  itself.  In  the  case  under  consideration  one 
of  the  lines  of  force,  namely  u  =  0,  does  intersect  itself,  having  a  double  point 
at  the  origin.  In  order  that  a  double  point  may  exist  the  partial  derivatives  of 
u  with  respect  to  x  and  y  must  vanish.*  These  partial  derivatives  are  the  com- 
ponents of  the  force  acting,  and  since  both  are  zero  there  can  be  no  force  at  such 
a  point.  For  this  reason  such  a  p<5int  is  called  a  point  of  equllibrium.t  In  the 
case  of  an  irrotational  fluid  motion  the  components  of  the  velocity  are  zero  at  a 
point  of  equilibrium  and  hence  no  flow  takes  place  at  such  a  point.  The  same 
configiu-ation  occurs  in  the  discussion  of  the  colored  rings  in  biaxial  crystals  due 
to  the  interference  of  polarized  light. 

By  means  of  the  logarithmic  fmiction,  we  may  express  the  more 
general  case  of  any  number  of  sources  and  any  number  of  sinks, 
each  having  a  given  strength.  Suppose  we  have  a  source  at  each  of 
the  points  ai,  a2,  .  .  .  ,  ««  having  strengths  of  A;i,  A:2,  .  .  .  ,  /fcn, 
respectively.  Let  there  be  a  sink  at  each  of  the  points  jSi,  /32,  .  .  .  ,  /3m, 
each  of  strength  Xi,  X2,  .  .  .  ,  Xm,  respectively.  Since  the  sinks  are 
to  be  considered  as  negative  sources,  the  corresponding  factors 
appear  in  the  denominator  of  the  function  of  which  the  logarithm 
is  to  be  taken.     The  corresponding  function  is  then 

,      {z  —  ai)*<z  —  02)*'  ...  (2  -  «„)*» 
10  =  log 


(2  -   ^i)^<Z  -  /32)^'     ...     (0  -  ^mf 


As  a  special  case  which  presents  some  interest,  let  us  consider  the  fimction 

(2  -  1)» 


w  =  log 


2  +  1 


The  function  w  determines,  for  example,  the  equipotential  lines  and  the  lines 
of  force  in  a  magnetic  field  about  two  parallel  conductors  in  which  the  electric 
current  is  passing  in  opposite  directions  in  the  two  and  is  twice  as  strong  in  the 
one  as  in  the  other.  The  wires  pierce  the  complex  plane  at  A  =  (1,  0)  and 
B  =  (  —  1,  0).  The  wire  through  A  carries  a  current  twice  as  strong  as  the  one 
through  B. 

In  order  to  obtain  the  two  systems  of  conjugate  curves  given  by  the  function, 
put 

z  —  1  =  pie**i,         2  +  1  = 


*  See  Townsend  and  Goodenough,  First  Course  in  Calculus,  p.  370. 
t  See  Maxwell,  Electricity,  Vol.  I,  Chap.  VI;  Jeans,  Electricity  and  Magne- 
tism, p.  59;   Lamb,  Hydrodynamics,  p.  17. 


142  MAPPING,  ELEMENTARY  FUNCTIONS  [Chap.  IV. 

we  have 

XD  =  log^^^  =  log^  +  loge<(»«i-»«) 

Ol&"*  Pi 


log^+t(2  0i-e2). 
Pt 


Henoe,  we  obtain 

u  =  log 

Pi 


M  =  log  — ,         v  =  201-0,. 


For  u  =  c,  we  have 


Pt 


Pi' 


or  p,=^  =  ii[pi^        K>0.  (12) 

For  the  orthogonal  system,  we  have 

0  =  201-02.  (13) 

To  plot  any  one  of  the  system  of  curves  represented  by  (12),  give  K  an  assigned 
value  and  give  to  pi  any  convenient  succession  of  values.  Compute  the  corre- 
sponding values  of  P2  by  means  of  (12).  With  z  =  + 1  as  a  center  and  the 
assumed  values  of  pi  as  radii,  draw  circles.  Likewise,  with  z  =  —  1  as  a  center 
and  the  computed  values  of  p2  as  radii  draw  circles.  The  intersections  of  corre- 
sponding circles  give  points  on  the  required  curve. 

To  plot  a  curve  belonging  to  the  system  given  by  (13),  give  to  c  any  assigned 
value  and  from  the  points  z  =  +l,z  =  — 1,  and  draw  lines  making  angles  6i  and 
02  =  2  01  —  c,  respectively,  with  the  axis  of  reals.  The  intersection  of  corre- 
8p>onding  Unes  gives  points  on  the  required  ciurve.  The  general  form  of  the  two 
systems  of  curves  is  shown  in  Fig.  62. 

To  determine  the  double  points  of  the  lines  of  force,  that  is,  the  points  of  equi- 
librium, we  have 

,    pi^    ,      (x  -  ly  +  2/2 

M  =  log  ^—  =  log 


PS  V(a;  +  l)2-f  y» 

The  double  points  are  given  by  putting  partial  derivatives  of  u  with  respect  to 
X  and  y  equal  to  zero  and  solving  the  two  resulting  equations  for  x  and  y.  We 
have  then  to  solve  the  equations 

9^^  d  (x-iy  +  j/^    ^     2  (a;  -  1)  a;  -H         ^ 

dx      dx'"^V(7+IFT?      ix-iy  +  y^      (x  +  iy  +  y^      "'       ^'■^^ 


2(x- 

1) 

(z  -  ly 

+  y' 

2y 

^^  ^  w    (x-iy+jf    ^  2y y  ^ 

dy      dj/'°«V(x-Hl)s-h2/2       ix-iy  +  y'      {x  +  iy+y^      "'      ^'^^ 

Equations  (14)  and  (15)  are  satisfied  simultaneously  by  the  values  y  =  0,x  =  —3. 
These  values  are  therefore  the  coordinates  of  the  point  C  of  equilibrium.     To 


Art.  30.] 


THE  FUNCTION  LOG  Z 


143 


determine  which  one  of  the  lines  u  =  c  maps  into  the  particulax  curve  having  a 
double  point  at  (  —  3,  0),  we  substitute  the  values  x  =  —  3,  t/  =  Oin  (12)  and 
determine  the  corresponding  value  of  c.     This  substitution  gives 

^'^  =  64,        or        c  =  log  8; 
that  is,  the  potential  function  has  at  each  point  of  this  curve  the  value  log  8. 


Fig.  62. 


The  distribution  of  matter  being  confined  to  the  plane,  the  intensity  of  the 
force  at  any  point  per  unit  of  strength  is  equal  to  the  reciprocal  of  the  distance.* 
Hence,  in  order  that  C  shall  be  a  point  of  equilibrium,  it  follows  from  the  laws  of 
physics  that  C  must  lie  on  the  X-axis  and  that  we  must  have 

AC      BC 

It  will  be  seen  that  this  equation  gives  the  same  values  of  the  coordinates  of  C 
as  those  already  obtained. 

*  See  Wangerin,  Theorie  des  Potentials  und  der  Kugdfwnktionen,  Vol.  I,  pp. 
135-137. 


144  MAPPING,  ELEMENTARY  FUNCTIONS  [Chap.  IV. 

31.  Trigonometric  Functions.  The  definition  of  the  various 
trigonometric  functions  may  be  made  to  depend  upon  the  expo- 
nential function  e  already  defined.  From  the  definition  of  e^  it 
was  shown  that 

e^=  COS0  +  isin0, 
whence 

e~**  =  cos  0  —  i  sin  0, 

where  B  in  both  cases  is  real.    Solving  these  equations  for  sin0, 
cos  5,  we  get 

e'*  —  e~ 


sin0  = 


COS0  = 


2i      ' 


In  a  similar  manner,  we  shall  now  define  sin  «,  cos  z  in  terms  of  the 
exponential  function  e*,  by  putting 

€**  —  e-** 
sm ;?  =  — J7-: 1 

COS  «  = 5 • 

Since  the  function  e*  is  analytic,  it  follows  that  sin  2,  cos  z  are 
also  analytic  functions.  Moreover,  sin  x  and  cos  x  appear  as  special 
cases  of  the  sine  and  cosine  of  the  complex  variable  z. 

The  trigonometric  functions  of  a  complex  variable  satisfy  the  same 
trigonometric  identities  as  the  corresponding  functions  of  real  vari- 
ables. We  may  show,  for  example,  that  the  following  relation 
holds 

sin  (zi  +  J82)  =  sin  Z\  cos  22  +  cos  Zi  sin  Zi. 
We  have 

sm  Z\  cos  02  +  cos  01  sm  02  =  -^^ ?r^-Si 

2i'2 

(e'*'  +  e-^O  (e»*«  —  e"*^')      2  e'(^i+«2)  —  2  e-»(«i+«s) 


2.2t  .  Ai 

g«(«l+2j)  _   g-t(2i+«i) 

—  sin  (01  +  22).  • 

The  remaining  trigonometric  identities  may  be  established  in  a  similar 
manner.     While  the  fundamental  identity 

cos^  z  +  sin'^  0  =  1 


Abt.  31.]  .        TRIGONOMETRIC  FUNCTIONS  145 

holds  for  complex  as  well  as  real  values  of  z,  it  follows  from  the  defi- 
nitions of  sin  z,  cos  z,  that  by  the  proper  choice  of  the  complex 
variable  z  either  of  the  functions  sin  z  and  cos  z  can  be  made  greater 
than  unity  in  absolute  values,  thus  differing  in  this  respect  from  the 
case  where  the  variable  is  real. 

Since  e"  has  the  period  2  tt,  it  follows  from  the  definition  of  sin  z 
that  it  likewise  has  the  period  2  tt;  for,  we  have 

gi(2+2x)_  g-i(2+2x) 

sin(2  +  27r)=^ 2^ 

_  e"  —  e-« 
~       2i 
=  sinz. 

In  a  similar  manner  it  may  be  shown  that  cos  z  has  the  period 
2  7r;  that  is,  that 

cos  (2  +  2  tt)  =  COS  z. 

The  remaining  trigonometric  functions  are  periodic,  having  the  same 

periodicity  as  the  corresponding  functions  of  a  real  variable.     As 

we  shall  see  the  lines  that  limit  the  fundamental  region  of  sin  z  and 

cos  z  are  parallel  to  the  F-axis,  while  the  fundamental  region  for  e* 

is  bounded  by  lines  parallel  to  the  axis  of  reals.     This  difference  is  a 

consequence  of  inserting  the  factor  i  before  z  in  the  definition  of 

sin  z,  cos  z.     It  is  to  be  noted  also  that  for  the  exponential  functions 

e^,  e"  the  region  of  periodicity  is  identical  with  the  fundamental 

region;  that  is  to  say,  no  two  points  z  iu  one  of  the  strips  defining 

the  region  of  periodicity  gives  the  same  value  of  the  function.     In 

the  case  of  sin  z  and  cos  z  the  situation  is  different  and  each  of  these 

functions  has  the  same  value  for  two  different  values  of  z  in  the  strip 

defining  the  region  of  periodicity.     For  example,  if  we  substitute 

jr  —  z  for  2  in 

e"  —  e-^' 
smz  = 


we  have  sin  (x  —  z)  = 


2i      ' 

g»(«— 2)  Q—i{x—z) 

2i 

"  ~2i 


Remembering  that 


e''^  =  cosTT  +  isinTT  =  —  1, 

,-ix   _   (,Qg  T^  —  i  sill  TT  =    —  1, 


146 


MAPPING,  ELEMENTARY  FUNCTIONS 


[Chap.  IV. 


we  have 


sin  (t  —  «)  = 


—  e-"  +  e' 


2i 


=  sin  2. 


Moreover,  the  points  representing  z  and  t  —  z  both  lie  within  the 
region  of  periodicity  —  x  <  x  =  tt,  as  shown  in  Fig.  63,  provided  the 
real  part  of  z  is  greater  than  zero  and  not  greater  than  r.     This  result 


''t 


!!l 

•   I    ■ 


i    i 


iiiiiiii 


...ji-i-j.-i.-i— I 

g!,q:"p"|,'?[>Tn"  j^ 


kTv'.a'.  b'\cy\o\  ^\'fM^''\'^\ 


Fig.  63. 


Fig.  64. 


shows  that  the  region  of  periodicity  of  sin  z  can  not  be  taken  as  a 
fundamental  region  of  that  function. 
Again,  if  in 

e"  +  e-« 


cos  2  = 


we  replace  —2  by  2,  we  have  the  same  function  as  before.  Hence,  we 
may  write 

cos  (  — 2)   =  cos  2. 

If  2  is  represented  by  a  point  in  the  upper  right-hand  portion  of  the 
strip  of  periodicity,  as  shown  in  Fig.  64,  then  —2  is  a  point  in  the 
lower  left-hand  portion  as  shown.  The  two  points  lie  symmetrically 
with  respect  to  the  origin.  This  fact  shows  that  the  region  of  peri- 
odicity —T<x  =  T  does  not  answer  the  purpose  of  a  fundamental 
region  for  cos  2. 

Neither  the  sine  nor  the  cosine  can  have  the  same  value  for  mere 
than  two  values  of  2  in  the  same  periodic  strip.  For  example,  for 
the  cosine,  we  have 

e"  +  e-" 


cos  2  = 


2 

?^*'  -j-  1 
2e"    ' 


(1) 


Art.  31.]  TRIGONOMETRIC   FUNCTIONS  147 

which  is  of  the  second  degree  in  e";  and  hence  if  we  put  cos  z  equal 
to  a  constant  there  are  but  two  values  of  e"  and  hence  of  z  that  sat- 
isfy this  equation. 

As  we  have  already  seen,  the  regions  of  periodicity  can  not  always 
be  taken  as  the  fundamental  region.  We  shall  show  that  the  region 
bounded  by  the  lines  re  =  0,  x  =  -w  may  be  taken  as  the  funda- 
mental region  for  cos  z.  In  this  connection,  we  shall  consider  the 
mapping  of  the  Z-plane  upon  the  W-plane  by  means  of  the  relation 
w  =  cos  z.    We  have 

.        e''-\-e-'' 
cosz  =  w  =  u-{-iv  = ^ 

=  2 

~  2 

e~''  e^ 

=  -^  (cos X  +  i  sin x)  -{-  -^  (cos x  —  isiax) 

=  -^ COSX-h* o ^^X'     -<l*oii^^X     t^ 

Hence,  we  may  write 

u  =  — ^—  cos  X,         V  =  — ^ —  sm  X.  (2) 


For  a;  =  0,  we  obtain 


u  =  — X —  ,        V  =  0. 


If  ?/  =  0,  we  have  u  equal  to  +1;  for  y  $  0,  we  get  m  >  1.     Hence, 
the  axis  of  imaginaries  maps  into  that  portion  of  the  U-axis  that 
lies  to  the  right  of  w  =  1,  as  shown  in  Fig.  65. 
For  a;  =  TT,  we  have 

e-y  +  e«  „ 

u= 2 '        ^  =  0» 

and  the  line  x  =  t  maps  into  that  portion  of  the  negative  C/-axis 
that  lies  to  the  left  of  the  point  —1.  If  0  <  x  <  tt,  we  have  for  a 
positive  value  oi  y  a  corresponding  negative  value  of  v;  for,  the 
value  of  sin  x  is  positive,  while  the  factor 

e-y  —  e" 


148 


MAPPING,   ELEMENTARY  FUNCTIONS 


[Chap.  IV. 


is  negative.    In  a  similar  way,  if  y  is  negative  and  0  <  x  <  ir,  the 
corresponding  point  in  the  TF-plane  Hes  above  the  axis  of  reals. 

It  will  be  seen  upon  inspection  that  for  x  =  x,  y  =  0,  the  value  of 

w  is  zero.     As  x  varies  from  0  to  x,  y  remaining  zero,  w  takes  all  of 


Fig.  65. 


Fig.  66. 


the  values  represented  by  points  on  the  real  axis  between  +1  and 
—  1 ;  for,  in  this  case  we  have  from  (2) 

u  =  cosx,        y  =  0. 

Lines  parallel  to  the  X-axis  map  into  ellipses  (Fig.  65)  having  the 
points  ±1  as  the  common  foci.  We  may  show  this  as  follows. 
From  (2)  we  get 

2m  .  -2v 


cosx  = 


6"  +  e-»' ' 


smx  = 


6"  —  e~" 


Squaring  both  members  of  these  equations  and  adding,  we  have 

For  y  =  c  this  equation  is  that  of  an  ellipse.     For  various  values  of 
c  we  obtain  a  system  of  ellipses  having  the  common  foci  +1,  —1. 

The  lines  parallel  to  the  F-axis,  that  is  x  =  c,  map  into  hyperbolas 
having  the  foci  ±1.  To  get  the  equations  of  these  hyperbolas,  we 
divide  the  members  of  the  j5rst  equation  in  (2)  by  cos  x  and  those  of 
the  second  by  sin  x,  thus  obtaining 


u 
cos  a; 


e-^  +  e" 


V 

sinx 


e-y  —  e" 


Abt.  31.]  TRIGONOMETRIC  FUNCTIONS  149 

Squaring  these  results,  we  have 


\cos  xj  4  \sin  x/ 


e-iy  _  2  +  e2y 


4 

Subtracting  the  second  of  these  equations  from  the  first,  we  obtain 


\cos  xJ       Vsm  xj        ' 


which  for  constant  values  of  x  is  the  equation  required. 

The  region  bounded  by  the  Hnes  x  =  0,  x  =  t  maps  into  the  entire 
W-plane  and  may  therefore  be  taken  as  the  fundamental  region  for 
cos  z.  Any  region  bounded  by  the  lines  x  =  kir,  x  =  {k  -{-  1)  ir, 
k  =  •  •  •,  —  3,  —  2,  —  1,  0,  +1,  +2,  +3,  .  .  .  answers  equally  well 
as  a  fundamental  region. 

The  corresponding  regions  in  the  two  planes  are  indicated  by  the 
letters  a,  6,  c  .  .  .  and  a',  6',  c',  .  .  .  ,  Figs.  65  and  66. 

The  configuration  in  the  TT-plane  (Fig.  65)  gives  us  a  method  of 
determining  cos  z  by  graphical  methods.  For  example,  let  2  ^  a  +  ih 
be  any  point  in  the  Z-plane.  Suppose  the  parallels  to  the  axes  map 
into  the  particular  ellipse  and  hyperbola  shown;  then  cos  z  is  repre- 
sented by  the  intersection  of  the  curves  as  indicated. 

From  the  definitions  of  sin  z  and  cos  z,  we  can  readily  obtain  ex- 
pressions for  the  other  trigonometric  functions  in  terms  of  the  ex- 
ponential function. 

For  example,  we  have 

sins  ^         cos  2     ^ 

tan  z  = ,        cot  z  =  -. — ,  etc. 

cos  z  sm  z 

These  functions  are  holomorphic  in  that  portion  of  the  finite  plane 
for  which  they  are  defined.  The  first  of  these  functions,  tan  z,  is 
undefined  for  those  values  of  z  for  which  cos  z  vanishes.  From  the 
map  of  cos  z  upon  the  TF-plane,  it  will  be  seen  that  cos  z  is  equal 
to  zero  only  for  the  real  values  \  yi 

z  =  ^  ±  A;r,        fc  =  0,  1,  2,  .  .  .  . 

In  a  similar  manner,  it  may  be  shown  that  cot  z  is  undefined  for 
those  values  of  z  for  which  sin  z  vanishes,  that  is  for  the  real  values 

z=  ±.k-K,        A;  =  0,  1,  2,  ...  . 
The  trigonometric  functions  are  therefore  analytic  functions. 


150  MAPPING,   ELEMENTARY  FUNCTIONS  (Chap.  IV. 

32.  Hyperbolic  Functions.     As  in  the  case  of  circular  functions, 
we  shall  first  define  the  functions 

w  =  sinh  z,        w  '=■  cosh  2, 

and  from  these  definitions  deduce  the  remaining  functions  by  means 
of  the  relations, 

^     .  sinh  z  ^,  1  .1 

taiih0  =  — :—■>        coth«=; — r— ,       secn«  =  — r— » 
cosh  z '  tanh  z  cosh  z 

cosech  z  = 


sinh« 


We  now  define  sinh  z,  cosh  z  in  terms  of  the  exponential  function  as 
follows: 

smh»= — 5 — ,       cosh2!  = — .  (1) 

By  comparing  these  definitions  with  those  of  the  sine  and  cosine,  it 
will  be  seen  at  once  that 

sinh  z  =  —  i  sin  iz,        cosh  z  =  cos  iz.  (2) 

The  following  useful  identities  follow  at  once  from  the  definitions 
given. 

cosh*^  z  —  sinh^  «  =  1, 

sech^  z  +  tanh** »  =  1» 

coth®  z  —  cosech^  »  =  1. 


To  deduce  the  first  relation,  we  have 

cosh^g  —  sinh^^  =  ( — - — j  —  / — - — j 


+  2  +  6-2*      c2'  -  2  +  e-' 


2  s 


=   1. 

The  rest  of  the  above  identities  may  be  deduced  in  a  similar  manner. 

The  hyperbolic  functions  are  analytic  functions,  since  e^  is  an 
analytic  function.  Moreover  these  functions  are  periodic;  for,  as 
we  have  seen  the  function  e"  is  periodic. 

The  formulas  for  hyperboUc  functions  of  real  variables  may  be 


Art.  32.1  HYPERBOLIC   FUNCTIONS  151 

extended  without  change  to  complex  variables.     We  have,  for  ex- 
ample, 

cosh  (zi  +  22)  =  cos  i  {zi  +  02) 

=  cos  izi  cos  iz2  —  sin  izi  sin  izi 
=  cosh  2i  cosh  Z2  +  sinh  Zi  sinh  Z2. 

Hyperbolic  functions  of  real  variables  may  often  be  conveniently 
used  to  express  the  trigonometric  functions  of  a  complex  variable  in 
the  form 

u{x,  y)  +  iv{x,  y). 
We  have,  for  example, 

sin  2  =  sin  {x  +  iy)  =  sin  x  •  cos  iy  +  cos  x  •  sin  iy 

=  sin  X  •  cosh  y  ■{-  i  cos  x  •  sinh  y, 
whence  w  =  sin  a;  cosh  y,     v  =  cos  x  sinh  y. 

Similarly  cos  z  =  cos  x  cosh  y  —  i  sin  x  sinh  y, 

and  u  =  cos  x  cosh  y,     y  =  —  sin  x  sinh  y. 

To  express  tan  z  in  the  form  u  +  iv,  we  have 
sin  2      sin  (x  +  i?/) 


tan  2  = 


cos  2      cos  (x  +  iy) 
sin  (x  +  iy)  cos  (x  —  it/) 


cos  (x  +  iy)  cos  (x  —  iy) 

_  sin2x  +  sin  2%  _  sin  2  x  +  *  sinh  2  y 

\  ^      cos  2  X  +  cos  2  iy       cos  2  x  +  cosh  2  y  ' 

,                                    sin  2  X  sinh  2  w 

whence  w  = ?; — ; r-^r— ,         y  = 


cos  2  X  +  cosh  2  y '  cos  2x4-  cosh  2  7/ 

We  shall  now  map  the  Z-plane  upon  the  TF-plane  by  means  of  the 
relation 

w  =  cosh  2. 

The  results  of  this  mapping  can  be  readily  deduced  from  those 
obtained  in  mapping  by  means  of  the  relation  w  =  cos  2;  for, 
it  will  be  seen  from  (2)  that  we  have  w  =  cosh  z  if  in  «;  =  cos  z, 
2  is  replaced  by  iz.  Hence,  to  map  any  configuration  from  the 
Z-plane  to  the  TT-plane  by  means  of  the  relation  w  =  cosh  2,  all 
that  is  necessary  is  first  to  map  the  given  configuration  from  the 
Z-plane  to  an  auxiliary  Z'-plane  by  means  of  the  relation  2'  =  iz, 
which  merely  rotates  each  point  of  the  complex  plane  through  a 

positive  angle  5,  and  then  to  map  the  resulting  configuration  from 


152  MAPPING,  ELEMENTARY  FUNCTIONS  [Chap.  IV. 

the  Z'-plane  to  the  TF-plane  by  means  of  the  relation  w  =  cos  z'. 
The  region  in  the  Z'-plane  bounded  by  the  Hnes  x'  =  0,  x'  =  r  may 
be  regarded  as  the  fundamental  region  for  the  function  w  =  cos  z'. 
This  region  corresponds  to  the  region  in  the  Z-plane  bounded  by  the 
lines  y  =  0,  y  =  —  TT,  which  may  therefore  be  taken  as  the  funda- 
mental region  for  the  function  w  =  cosh  z.  As  may  be  seen,  any 
one  of  the  regions  bounded  by  the  lines 

y  =  kir,     y  =  {k-l)ir,    A;  =  •  •  •  ,  -2,  -1,  0,  1,  2,  .  .  . 

can  be  used  as  a  fundamental  region. 

A  system  of  lines  in  the  Z-plane  parallel  to  either  of  the  coordi- 
nate axes  maps  by  means  of  the  relations 

w  =  cosh  2,     w  =  cos  z 

,M(into  a  system  of  straight  lines  in  the  TF-plane  which  are  likewise  par- 

Ij  allel  to  the  coordinate  axes.     The  lines  that  map  in  the  one  case  into 

ellipses  map  in  the  other  case  into  hyperbolas  and  conversely.     This 

result  is  verified  by  a  comparison  of  the  equations  for  u  and  v  in  the 

two  cases.     We  obtain  from  w  =  u-\-  iv  =  cosh  z 

u  =  cosh  X  cos  y  =  — x — -  cos  y, 

V  =  sinh  xainy  = ^ —  sin  y, 

which  are  the  same  equations  as  those  obtained  from  w  =  cos  z, 
Art.  31,  except  that  x  is  replaced  by  y  and  y  by  —x.  In  other  words 
by  the  change  from  cos  z  to  cosh  z  the  lines  of  level  and  the  lines  of 
slope  are  interchanged. 

In  a  similar  manner  we  may  establish  relations  between  the  maps 
obtained  by  means  of  the  remaining  circular  functions  and  the  corre- 
sponding hyperbolic  functions.  From  (2)  it  will  be  seen  that  the 
introduction  of  the  factor  i  enables  us  to  express  any  hyperbolic 
function  in  terms  of  the  corresponding  circular  function.  Conse- 
quently, it  follows  that  the  special  significance  of  hyperbolic  func- 
tions is  confined  to  functions  of  a  real  variable. 

Because  of  the  similarity  of  the  configurations  obtained  by  mapping  the  lines 
X  =  c,  y  =  c,  by  means  of  the  relations  w  =  cos  z  and  w  =  cosh  z,  it  is  to  be 
expected  that  similar  applications  may  be  made  in  theoretical  physics.  If  we 
have  the  case  of  a  liquid  flowing  about  an  elliptic  cylinder  whose  intersection  by 
the  complex  plane  is  an  ellipse  having  its  foci  at  —1  and  +1,  respectively,  then 
the  ellipses,  Fig.  65,  are  the  lines  of  flow  and  the  hyperbolas  are  the  lines  of  equal 


Art.  32.]  EXERCISES  153 

velocity-potential.  As  a  limiting  case  we  have  the  flow  of  a  liquid  about  a  thin 
plate  joining  the  points  +1  and  —1.*  If  that  portion  of  the  positive  real  axis 
lying  to  the  right  of  +1  be  regarded  as  a  line  source  and  that  portion  of  the 
negative  real  axis  lying  to  the  left  of  —  1  be  taken  as  a  sink  the  ellipses  are  again 
the  Unes  of  flow  and  the  hyperbolas  equipotential  lines.  If,  however,  the  line 
joining  +1  and  —1  is  regarded  as  a  source,  then  the  hyperbolas  are  the  lines  of 
flow  and  the  ellipses  are  the  equipotential  lines. 

The  definitions  of  the  transcendental  functions  thus  far  discussed 

have  been  based  upon  the  definition  of  e%  which  in  turn  was  defined 

in  terms  of  known  functions  of  real  variables.     Other  methods  of 

procedure  could  have  been  employed.     For  example,  the  logarithm 

C'dz 
of  z  could  have  been,  and  often  is,  defined  as  the  integral     /    — 

Ji   z 

From  this  definition  the  properties  of  a  logarithm  can  be  readily 

developed.     Then  e'  may  be  defined  as  the  inverse  function  of  log  z, 

and  the  remaining  functions  can  be  defined  as  in  the  text.     The 

other  transcendental  function  which  we  have  given  may  also  be 

defined  by  means  of  integrals;  for  example,  we  may  make  use  of  the 

following  relations  as  definitions 

A    dz  .  C       dz 

arc  tan  2=    I    - — ; — r,        arcsm0=    f       ,  , 

Jo  1  +  Z^'  Jo   Vl  -z" 

upon  which  the  definitions  of  the  remaining  functions  discussed  in 
the  text  may  be  based. 

EXERCISES 

1.  Discuss  the  conjugate  functions  determined  by  the  relation 

t«2  =  2  +  1. 

Plot  the  projections  upon  the  XF-plane  of  the  lines  of  level  and  lines  of  slope. 

2.  Discuss  the  mapping  upon  the  TF-plane  of  a  system  of  concentric  circles 
about  the  origin  in  the  Z-plane,  by  means  of  the  relation 

IP  =  3  z3  +  5. 

3.  Show  that  the  function 

z  —  r\  cos  \t  +  i  sin  \i  \ , 

whfere  t  is  the  independent  variable  representing  time  and  where  r  and  X  are  real 
constants,  represents  a  movement  of  the  z-point  such  that  the  velocity  v  of  the 
z-point  is  constant  in  magnitude  but  varying  in  direction,  and  such  that  the 
acceleration  of  the  2-point  is  always  directed  toward  the  origin  and  is  constant 

in  magnitude  and  equal  to  — ,  o-  being  the  absolute  value  of  v. 
r 

*  See  Lamb,  Hydrodynamics,  3^  Ed.,  p.  69;  Webster,  Electricity  and  Magnet- 
ism, p.  319. 


154  MAPPING,   ELEMENTARY  FUNCTIONS  [Chap.  IV. 

4.  Given  w  =  e*.  Suppose  that  the  a-p>oint  has  a  constant  real  velocity  <r 
along  the  line  y  =  <f>  in  the  Z-plane.  Show,  by  differentiation,  that  the  corre- 
sponding point  of  the  PT-plane  moves  along  the  straight  line  u  =  v  '  cot  <^  with  a 
varying  speed. 

6.  Any  straight  line  through  the  origin  making  an  angle  different  from  zero 
with  the  X-axis  crosses  an  infinite  number  of  fundamental  regions  of  the  function 
w  =  e*.  Explain  the  fact  that  such  a  line  maps  into  a  single  continuous  curve 
in  the  TT-plane. 

6.  Given  it;=flH — J  ,n  =  2,  3,  .  .  .  .     Determine  fundamental  regions 

for  this  function  for  the  various  values  of  n  and  show  how  we  may  obtain  a 
fundamental  region  for  it;  =  e*  as  the  limiting  case. 

7.  Show  that  e*  is  an  automorphic  function. 

8.  Construct  the  map  of  the  function  w  =  sinz  similar  to  the  map  of  cos  z 
shown  in  Figs.  65,  66. 

9.  Making  use  of  Figs.  65,  66,  and  the  figiires  obtained  for  the  function 
tc  =  sin  2,  construct  the  corresponding  figures  for  the  functions  w  =  cosh  z, 
w  =  sinh  z. 

( 10.   Show  that  Dj  sin  z  =  cos  z,  Dz  sinh  z  =  cosh  z. 
t11.   Show  that  sin  2  z  =  2  sin  z  cos  z;  sinh  2  2  =  2  sinh  z  cosh  z. 

12.  Show  that  for  w  =  sinh  z,  we  have  u  =  sinh  x  cos  y,v  =  cosh  x  sin  y. 

13.  Show  that  for  w  —  cosh  z,  we  have  u  =  cosh  x  cos  y,v  =  sinh  x  sin  y. 

14.  Prove  that 
sinh  X  cosh  a;  + 1  sin  j/  cos  y 


tanhz  = 


cos*  y  cosh*  X  +  sin*  y  sinh*  x 


16.   Discuss  the  mapping  of  orthogonal  systems  of  straight  lines  parallel  to 
the  axes  in  the  TF-plane  upon  the  Z-plane  by  means  of  the  relation 

w  =  log  (z  —  1)  (z  +  1)  (z  —  i). 

Discuss  the  possible  applications  in  theoretical  physics. 
16.   Discuss  the  function 

w  =  log -J — . 


and  point  out  possible  applications  as  suggested  by  the  map  in  the  Z-plane  of 
the  lines  u  =  c,  v  =  c.    Locate  the  points  of  equiUbrium,  if  such  exist. 

17.   Suppose  a  system  of  equipotential  curves  to  be  given  by  the  confocal 
ellipses 

Show  that  the  lines  of  flow  are  the  confocal  hjrperbolas 

^X  +  ^  =  l'      -a*<X^-6*. 


Art.  32.]  EXERCISES  155 

18.  Show,  by  the  method  of  Ex.  3,  that  for  uniform  motion  along  any  curve,, 
the  acceleration  is  always  directed  toward  the  centef  of  curvature  and  in  magni- 
tude is  equal  to  ^ 

radius  of  curvature  ^  ^ 

where  <r  is  the  speed  in  the  path, 

19.  Show  that  for  non-uniform  motion  in  any  curve,  the  component  of  the 
acceleration  normal  to  the  path  is 

a^ 

radius  of  curvature  j 

where  a  is  the  varying  speed  in  the  path, 

20.  The  logarithmic  function  and  the  hyperbolic  fimctions  have  been  defined 
in  terms  of  e".    By  means  of  these  definitions,  show  that 


w  + 1  8       w      ' 

z  =  log  — — -j-  =  2  arc  tanh  w.  TL  ''^^    cv  4-  / 

e  Cau 
point  P  =  ^ ,  ^  =  I 


21.  By  aid  of  the  Cauchy  integral  formula,  compute  the  value  of  sin  z  at  the  f2^^^^ 

1  ^ 


ic^- 


Z-T: 


UJ 


^  ^^ 


\- 


CHAPTER  V 

LINEAR  FRACTIONAL  TRANSFORMATIONS 

33.  Definition  of  linear  fractional  transformation.  In  several 
of  the  illustrative  examples  of  mapping  thus  far  considered,  the 
relation  between  w  and  z  is  such  that  a  portion  of  the  one  plane 
maps  into  the  whole  of  the  other  plane.  For  example,  in  the  case 
of  MJ  =  2*  one-half  of  the  Z-plane  maps  into  the  whole  of  the  TT-plane. 
To  each  point  in  the  TT-plane  there  correspond  then  two  points  in 
the  Z-plane,  symmetrically  situated  with  respect  to  the  origin.  We 
shall  now  consider  the  general  linear  algebraic  relation  between  w  and 
z.    This  relation  may  be  written  in  the  form 

w  =  ^^   ,    ,>  (1) 

70  +  5 

where  a,  /3,  7,  5  are  constants,  real  or  complex,  and  the  determinant 

a    /3 
7     h 

is  different  from  zero.  The  relation  (1)  between  w  and  z  differs  from 
those  mentioned  above  in  this  respect  that  to  each  value  of  z  there 
is,  with  a  convention  as  to  the  point  at  infinity  to  be  noted  in  the 
following  article,  one  and  only  one  value  of  w  and  vice  versa. 

In  our  discussion  of  mapping  (Chapter  IV),  we  examined  the 
relation  between  w  and  z  by  allowing  one  of  these  variables  to  de- 
scribe a  given  configuration  and  determining  the  corresponding 
configuration  described  by  the  other  variable.  The  one  configura- 
tion was  then  said  to  be  mapped  upon  the  other  configuration  by 
means  of  the  given  functional  relation.  In  a  similar  manner  a  given 
region  was  frequently  mapped  upon  another  region.  It  was  con- 
venient to  represent  the  i^-points  in  one  complex  plane  and  the 
2;-points  in  another,  employing  for  this  purpose  two  distinct  sets  of 
coordinate  axes,  one  in  each  plane.  In  the  present  chapter  we  shall 
study  the  particular  functional  relation  given  in  (1)  from  a  somewhat 
different  point  of  view.  We  shall  represent  both  the  z-points  and 
the  ly-points  in  the  same  plane  and  refer  them  to  the  same  set  of 

156 


Art.  34.]  POINT  AT  INFINITY  157 

coordinate  axes.  We  shall  attempt  to  find  a  path  along  which  any 
particular  2-point  may  be  regarded  as  moving  in  passing  into  the 
corresponding  w-point.  To  distinguish  this  process  from  that  of 
mapping  already  discussed,  we  shall  speak  of  it  as  a  transformation 
of  the  complex  plane.  Since  equation  (1)  expresses  w  as  a  linear  frac- 
tional function  of  z,  we  shall  call  the  transformation  to  be  discussed 
a  linear  fractional  transformation  of  the  complex  plane.  When  the 
relation  given  in  (1)  reduces  to  the  form  i«  =  az  +  /3,  we  shall  speak 
of  it  as  a  linear  transformation.  If  we  think  merely  of  the  results 
of  such  a  transformation,  we  may  say  that  a  given  configuration  is 
mapped  upon  the  resulting  configuration  by  means  of  a  linear  frac- 
tional transformation. 

34.  Point  at  infinity.  The  general  linear  fractional  relation 
given  in  (1)  of  the  last  article  fails  to  establish  a  complete  one-to-one 
correspondence  between  the  finite  points  of  the  complex  plane. 
For  example,  there  is  no  finite  value  of  w  corresponding  to  the  value 

2  = .     To  make  the  one-to-one  correspondence  complete,  it  is 

7 
customary  to  assign  an  ideal  point  to  the  complex  plane,  called  the 
point  at  infinity.  To  this  artificial  point  may  be  associated  the 
artificial  number  go  .  The  complex  plane  is  to  be  regarded  then  as 
closed  at  infinity;  that  is,  the  plane  is  to  be  considered  as  having  but 
a  single  point  at  infinity. 

We  set  up  a  correspondence  between  the  point  z  = and  the 

ideal   point  w  =  cc ,  and   likewise  between  the  ideal  point  z  =  co 

and  point  w  =  -.     The  one-to-one  correspondence  between  the  points 

7 
of  the  whole  complex  plane  by  means  of  the  general  linear  fractional 

relation  becomes  complete  by  the  aid  of  this  convention;    that  is, 

to  each  point  that  may  be  assigned  to  z,  there  is  one  and  only  one 

point  that  represents  the  corresponding  value  of  w,  and  conversely. 

This  convention  concerning  the  point  at  infinity  is  very  convenient 

in  other  connections.     When  we  speak  of  the  existence  of  the  limit 

L  f{z),  where  a  is  a  finite  number,  it  is  implied  that  the  function 

f{z)  has  the  same  limiting  value  as  z  approaches  a  through  all  pos- 
sible sets  of  values,  that  is,  a  limiting  value  that  is  independent  of 
any  changes  in  the  amplitude  of  z  —  a  as  the  modulus  o(  z  —  a  de- 
creases. Similarly,  a  function  f{z)  may  have  a  unique  limiting  value 
as  z  becomes  infinite  through  all  possible  sets  of  values;  that  is,  it 


158  LINEAR  FRACTIONAL  TRANSFORMATIONS         [Chap.  V. 

may  have  a  limiting  value  that  is  independent  of  any  changes  in  the 
amplitude  of  «  as  the  modulus  of  z  increases  beyond  all  finite  bounds. 
It  is  convenient  to  denote  this  limit  by  L  /(«)  and  to  speak  of  2  =  oo 

as  the  limiting  point  in  this  case. 

Let  us  now  consider  the  limiting  value  of  the  function 

W  =  —r^ 

72  +  5 

as  2  becomes  infinite  and  likewise  that  of  the  inverse  function 

-5MJ  +  /3 


as  w  becomes  infinite 


'''  —                       > 
7iy  —  a. 

.    We  have 

L  w  =  L   — r  =  -  , 

r=oo         «=oo  yz-\-  5      7 

w=oo         tc=oo  yw  —  a 

_8_ 

7 

and 


These  two  results  fully  justify,  in  so  far  as  the  general  linear  fractional 
function  is  concerned,  the  convention  introduced  concerning  the 
nature  of  the  complex  plane  at  infinity. 

If  we  think  of  the  values  of  2  as  represented  by  the  points  of  one 
complex  plane,  called  the  Z-plane,  and  the  values  of  w  as  represented 
by  the  points  of  another  complex  plane,  called  the  TF-plane,  then  of 
course  an  ideal  point  at  infinity  must  be  associated  with  each  plane. 

We  may  speak  of  the  neighborhood  of  the  point  at  infinity  just 
as  we  speak  of  the  neighborhood  of  any  finite  point.  By  such  a 
neighborhood  is  understood  the  set  of  points  exterior  to  any  closed 
curve,  for  example  the  set  of  points  exterior  to  a  large  circle  about 
the  origin  as  a  center.  We  say  also  that  a  given  region  contains  the 
point  at  infinity  if  it  consists  of  all  the  points  exterior  to  a  given 

closed  curve.     In  this  connection  the  substitution  2  =  —  is  of  special 

importance.  By  this  transformation  every  finite  point  except  the 
origin  goes  over  into  some  finite  point  of  the  plane.  The  neighbor- 
hood of  the  origin  corresponds  to  the  neighborhood  of  the  point  at 
infinity,  and  the  origin  itself  may  be  said  to  correspond  to  the  point 
at  infinity. 

This  relation  between  the  point  at  infinity  and  the  origin  affords 
a  convenient  method  of  investigating  the  properties  of  a  function 


Awre.  35-36.]  W  =  Z -\-  fi,   W  =  aZ  159 

f{z)  for  values  of  z  in  the  neighborhood  of  the  point  2  =  oo .    To  do 

so  we  replace  in  /(«)  the  independent  variable  2  by  -7  and  discuss  the 

properties  of  the  transformed  function  <!>{/)  in  the  neighborhood  of 
the  point  z'  =  0.     If  the  limit   L  <f>(z')  exists  and  is  equal  to  A,  then 

2'=0 

we  say  that  f{z)  has  the  value  A  at  the  point  at  infinity,  and  we 
write  A=    L    f(z)  =/(oo). 

If  <l){z')  is  contuiuous  for  2'  =  0,  we  say  that  f{z)  is  continuous  at 
infinity.  If  z'  =  0  is  a  regular  point  of  (t>{z'),  we  say  that  2  =  00  is 
a  regular  point  of  f{z).    As  2  becomes  infinite  the  function  f{z)  may 

also  become  infinite  and  in  such  a  manner  that  77-r  approaches  the 

/(z) 

limiting  value  zero.     We  say  then  that 

/(oo)  =  CX). 

35.  The  transformation  w  =  z  -\-  ^.  Before  discussing  the  gen- 
eral transformation  (1)  of  Art.  33,  we  shaU  consider  some  special 
cases  that  are  of  particular  importance  and  first  of  all  let  us  consider 
the  transformation 

w  =  z  +  j8. 

To  obtain  this  function  from  the  general  case,  put  a  =  5  =  1  and 
7  =  0.  This  relation  indicates  that  to  each  number  z  there  is  added 
another  number  /3.  From  the  geometric  interpretation  of  addition 
it  will  be  seen  that  the  z-points  are  transformed  into  the  correspond- 
ing ty-points  by  moving  each  2-pouit  in  the  direction  of  the  line 
joining  the  point  jS  with  the  origin  and  to  a  distance  equal  to  |  /3  |  . 
In  what  follows,  it  frequently  will  be  convenient  to  describe  a  trans- 
formation of  the  complex  plane  as  a  continuous  motion,  meaning 
thereby  that  all  of  the  points  of  the  plane  are  considered  as  moving 
continuously  from  their  initial  to  their  final  positions  along  a  system 
of  curves.  The  motion  just  described  is  called  a  translation.  It 
will  be  seen  that  a  translation  of  the  complex  plane  leaves  the  form 
and  size  of  any  configuration  unchanged;  that  is,  it  transforms  any 
curve  into  a  congruent  curve. 

36.  The  transformation  w  =  az.  As  already  stated  a  may  be 
a  real  number  or  a  complex  number  of  the  form 

a  =  p  (cos  6  -\-  i  sin  6). 


160 


LINEAR  FRACTIONAL  TRANSFORMATIONS        [Chap.  V. 


We  shall  understand  more  easily  the  full  significance  of  this  trans- 
formation by  considering  first  some  special  cases.  Let  us  suppose  for 
example  that  6  =  0;  that  is,  let  a  be  considered  as  a  real  positive 
number.  The  result  is  simply  a  multiplication  of  the  modulus  of  z 
by  the  number  p  =  \a\.  Geometrically,  this  special  form  of  the  gen- 
eral transformation  may  be  regarded  as  moving  every  point  along  the 
line  passing  through  the  given  point  and  the  origin,  that  is,  along  the 
half-ray  from  the  origin  on  which  it  lies.  The  point  moves  out  or  in 
along  this  half-ray  according  as  p  is  greater  than  or  less  than  unit5^ 
This  change  affects  every  point  of  the  complex  plane,  and  we  may  re- 
gard the  transformation  as  representing  a  motion  of  the  points  of  the 
plane.  We  shall  refer  to  such  a  motion  as  an  expansion  or  stretching. 
We  are  concerned  here  with  the  path  by  which  the  variable  point 
may  be  regarded  as  passing  from  its  initial  to  its  final  position,  rather 
than  the  velocity  with  which  this  motion  takes  place.  The  number 
p  is  called  the  modulus  of  expansion. 

The  significance  of  p  may  also  be  seen  from  a  consideration  of  the 
derivative.     As  we  have  seen,  |  DzW  \  gives  the  ratio  of  magnification 

that  takes  place  in  infinitesimal  ele- 
ments as  2  varies.    We  have 

\D.w\  =  \D.iaz)\  =  \a\=p; 

that  is,  any  configuration  in  the  com- 
plex plane  is  magnified  in  this  ratio. 
Suppose  that  we  have  a  system  of 
-^  concentric  circles  about  the  origin  and 
a  pencil  of  rays  passing  through  the 
origin.  Each  haK-ray  remains  un- 
changed as  a  whole,  although  any 
particular  point  upon  it  is  moved  out 
or  in  according  as  the  ratio  of  expan- 
sion is  greater  or  less  than  unity. 
By  this  transformation,  any  portion 
of  the  plane  inclosed  by  two  half-rays  and  two  concentric  circles  is 
transformed  into  another  portion  bounded  by  the  same  two  haK-rays 
and  the  concentric  circles  into  which  the  first  two  are  transformed; 
for  example,  the  region  (2),  Fig.  67,  goes  over  into  (3).  Each  dimen- 
sion of  the  region  has  been  multiplied  by  the  number  p. 

Suppose  we  now  allow  6  to  vary,  while  p  remains  constantly  equal 
to  one.    Then  by  the  laws  of  multiplication  already  established,  we 


Fig.  67. 


Art.  36.]  THE  TRANSFORMATION  W  =  aZ  161 

obtain  from  any  value  z  the  corresponding  value  of  w  by  adding  to 
the  amplitude  of  z  the  angle  d.  Inasmuch  as  p  =  1,  no  magnification 
takes  place  and  the  resulting  configuration  is  obtained  by  revolving 
each  point  z  about  the  origin  counter-clockwise  through  the  angle  6. 
Considered  from  the  standpoint  of  the  geometry  of  motion,  this 
transformation  may  be  regarded  as  a  rotation  of  points  of  the  plane. 
Such  a  motion  converts  the  region  (3),  for  example,  into  the  region 
(4),  as  indicated  in  Fig.  67.  The  concentric  circles  about  the  origin 
then  become  the  lines  of  motion.  Each  ray  is  converted  into  another 
ray  at  an  angular  distance  d  from  it. 

Let  us  now  consider  the  general  case  where  a  is  any  complex  num- 
ber. Both  p  and  6  may  have  any  constant  values.  We  have  then 
a  combination  of  the  two  special  cases  already  considered;  that  is, 
the  point  is  rotated  about  the  origin  through  the  angle  6  while  it  is 
at  the  same  time  moved  along  the  ray  on  which  it  is  rotated;  for,  if 

we  have 

a  =  p(cos  6  -\-  i  sin  6), 

z  =  r(cos  <}>  -\-  i  sin  (f>), 

then  by  multiplication  we  get 

w  =  rp(cos  0  +  0  +  2  sin  0  +  0) ; 
that  is 

mod  w  =  r  '  p,         amp  w  =  6  -\-  <l>.  (1) 

The  result  of  the  transformation  w  =  az  may  be  obtained,  as  we 
shall  now  show,  by  regarding  each  point  as  moving  along  a  logarithmic 
spiral  whose  asymptotic  point  is  at  the  origin.  For  this  reason  it  is 
convenient  to  describe  the  given  transformation  as  a  logarithmic 
spiral  motion  about  the  origin. 

Let 

w  =  r'(cos  0'  -\-  i  sin  </>') 

be  the  point  into  which  the  point 

z  =  r(cos  (f>  -\-  i  sin  0) 

is  mapped  by  means  of  the  given  transformation  w  =  az.  From  (1) 
we  then  have 

r'  =  r'p,        <t>'  =  <t>-^d- 

A  logarithmic  spiral  passing  through  z  may  be  written  in  the  form 

r  =  ce**. 


162  LINEAR  FRACTIONAL  TRANSFORMATIONS         [Chap.  V. 

In  order  that  this  same  curve  shall  also  pass  through  w,  it  is  sufficient 

that 

r'  =  ce**' 

that  is,  it  is  sufficient  that 

r'  =  T'  p=  re^, 
or 

^~   e 

Hence,  if  the  logarithmic  spiral  whose  equation  is 

r  =  ce'  (2) 

passes  through  the  point  z  it  also  passes  through  the  point  w  into 
which  2  is  mapped  by  the  given  transformation.  Consequently, 
the  given  z-point  may  be  regarded  as  passing  into  the  corresponding 
ly-point  by  a  motion  along  this  spiral. 

As  p  and  6  are  both  determined  by  a,  it  follows  that  a  determines 
the  particular  system  of  spirals  given  in  (2)  along  which  any  z-point 
may  move  into  the  corresponding  ty-point.  The  arbitrary  constant 
c  is  the  parameter  of  the  system,  and  to  each  point  z  there  corre- 
sponds one  and  only  one  value  of  c  and  hence  one  and  only  one 
logarithmic  spiral  of  the  system.  The  entire  plane  is  filled  by  this  .  i 
system  of  curves,  coiled  up  within  each  other.  y 

37.  The  transformation  t«>  =  as  +  p.  The  significance  of  this 
transformation  may  be  most  readily  seen  by  regarding  it  as  a  com- 
bination of  the  two  preceding  transformations.  Let  a  logarithmic 
spiral  motion  take  place  about  the  origin,  and  then  let  the  result  be 
translated  by  adding  the  number  /3.  Analytically,  this  result  is 
equivalent  to  introducing  the  auxiliary  variable  z',  defined  by  the 
equation  ^,  ^  ^^  (1) 

and  following  this  transformation  by  that  of 

w  =  z'  +  ^.  (2) 

The  given  transformation  is  therefore  equivalent  to  a  logarithmic 
spiral  motion,  that  is  a  rotation  and  a  stretching,  followed  by  a  trans- 
lation. 


Art.  37.]  THE  TRANSFORMATION   W  =  aZ -{- 0  163 

The  question  naturally  arises  as  to  whether  we  may  not  reverse 
these  two  operations;  namely,  whether  we  may  not  take  first  the 
translation  and  then  the  logarithmic  spiral  motion.  Analytically, 
the  relation  w  =  az  -\-  fi  may  be  obtained  by  first  introducing  the 
auxiliary  variable 

0"  =  2  +  ^  ,  (3) 

a 

and  then  putting 

W  =  az".  (4) 

The  amount  of  the  rotation  and  stretching,  that  is  the  extent  of  the 
logarithmic  spiral  motion,  given  by  (4)  and  (1),  is  determined  by  the 
complex  constant  a.  Since  the  value  of  a  is  the  same  in  both  equa- 
tions, the  motion  is  the  same.  The  translations  defined  by  (3)  and 
(2)  are,  however,  different.  Hence,  we  see  that  a  logarithmic  spiral 
motion  and  a  translation  are  processes  that  can  not  be  interchanged. 
As  we  have  seen,  the  processes  of  rotation  and  stretching  are  on  the 
other  hand  interchangeable  processes. 

Whenever  we  apply  the  transformation  w  =  ocz  -\-  fi,  there  is  one 
point  of  the  plane  that  remains  unchanged;  that  is,  there  is  one  in- 
variant point.  We  can  readily  locate  this  point,  since  in  this  case  the 
2:-point  is  identical  with  the  corresponding  w-point.  Let  us  therefore 
write 

z  =  az  -\- 13, 

and  from  this  equation  determine  the  value  of  z. 
Then,  for  a  9^  1,  we  have  as  the  invariant  point 

8 
z  = 


1  -a 


The  given  transformation  represents  a  logarithmic  spiral  motion 
about  this  invariant  point.  For,  referring  the  points  of  the  plane 
to  the  invariant  point  as  the  origin,  that  is  putting 


w  =  w'  +  =-^ ,         z  =  z'+      ^ 


we  have 


or 


1-a'  '    1-a' 

w'  =  az'.  (6) 


164 


LINEAR  FRACTIONAL  TRANSFORMATIONS         [Chap.  V. 


This  equation  represents  a  logarithmic  spiral  motion  about  the  new 
origin,  that  is  about  the  invariant  point 


z  = 


/3 


1  -a 


As  we  have  already  seen,  a  logarithmic  spiral  motion  converts  a 
given  configuration  into  a  similar  configuration.  The  amount  of 
rotation  that  takes  place  in  the  logarithmic  spiral  motion  represented 
by  the  transformation  w  =  az  -{-  ^  is  given  by  the  amplitude  of  a. 
The  magnification  that  takes  place  in  the  elements  of  the  configura- 
tion by  means  of  this  transformation  is  determined  by  |  a  ]  =  p; 
for,  we  have 

I  D,w;  I  =  I  D,{az  +  /3)  1  =  I  a  I  =  p. 

Since  all  elements  are  magnified  by  the  same  amount  and  otherwise 
the  configuration  remains  unchanged  except  in  position,  we  may 
conclude  that. the  general  linear  transformation 

w  =  az  -]-  ^ 

transforms  the  complex  plane  into  itself  in  such  a  manner  that  any 
given  configuration  is  converted  into  a  similar  configuration  whose 


position  is  determined  by  a  and  the  constant 


1  - 


and  whose 


relative  size  is  determined  by  |  a  |  alone. 

Conversely,  we  may  show  that  any  transformation  of  the  plane 
into  itself  which  preserves  the  similarity  of  the  figure  is  a  linear 
transformation  of  the  form  w  =  az  -\-  ^.  Suppose  that  we  have  given 
two  similar  plane  figures.  Since  a  linear  function  of  the  form  under 
discussion  has  two  arbitrary  constants,  a  function  of  this  kind  can 
always  be  found  that  will  transform  any  two  distinct  points  Zi,  02  of 
the  one  configuration  into  any  two  given  distinct  points,  say  the 
points  Wi,  Wi,  of  the  second  configuration  homologous  respectively  to 
Zi,  22.  For  the  determination  of  these  two  constants  we  have  the  two 
equations 

Wi  =  azi  +  /3, 

W2  =  aZ2-\-  fi, 
whence,  we  obtain 


Wi    1 

Zl   Wi 

W2    1 

,      ^  = 

22   W2 

2l    1 

2i    1 

Z2    1 

22    1 

Art.  37.] 


THE  TRANSFORMATION  W  =  aZ -]-  ^ 


165 


The  functional  relation  that  transforms  the  two  points  Zi,  z^  into  the 
two  points  wi,  Wi  is  therefore 


w  = 


Wi    1 

Zl   Wi 

Vh  1 

z  + 

22   WJ2 

zx  1 

2l    1 

22    1 

22    1 

(6) 


Wl 

1 

Vh 

1 

Zx 

1 

22 

1 

The  amount  of  rotation  and  stretching  that  takes  place  in  this  trans- 
formation is  determined  by 


Wx  —  Wz  , 

01  —  22   ' 


the  rotation  is  given  by  the  amphtude  of  this  ratio,  that  is  by 
amp  {wi  —  W2)  —  amp  (21  —  Z2),  while  the  modulus  of  this  quotient 
gives  the  ratio  of  magnification  of  the  element  Zi  —  22. 

In  addition  to  this  rotation  and  magnification,  the  transformation 
given  by  (6)  involves  a  translation  of  the  points  of  the  complex  plane. 
The  amount  and  direction  of  this  translation  is  determined  by  the 
quotient 


Zx 

w, 

22 

Wi 

Zx 

1 

22 

1 

Since  the  two  configurations  are  similar,  the  amount  of  rotation, 
stretching,  and  translation  necessary  to  transform  any  element  21—22 
into  its  corresponding  element  Wi  —  W2  will  also  transform  any  other 
element  into  its  corresponding  element.  The  required  transforma- 
tion is  fully  determined  when  the  values  of  a  and  /3  are  expressed  in 
terms  of  known  values,  and  consequently  equation  (6)  gives  the  trans- 
formation sought. 

If  it  is  known  that  there  exists  a  relation  between  w  and  2  which 
is  holomorphic  in  a  certain  portion  of  the  complex  plane  and  if  by 
means  of  this  functional  relation  a  given  configuration  is  transformed 
into  one  similar  to  it,  then  it  is  possible  to  show  by  a  consideration  of 
the  derivative  that  this  relation  is  linear.  As  already  pointed  out,  the 
ratio  of  magnification  that  takes  place  in  passing  from  the  Z-plane 
to  the  TT-plane  by  means  of  a  transformation  w  =  f(z)  is  given  by 
the  modulus  of  D^w,  while  the  amplitude  of  this  derivative  gives  the 


166  LINEAR  FRACTIONAL  TRANSFORMATIONS        [Chap.  V. 

rotation  that  takes  place.  In  the  case  under  consideration  both  the 
ratio  of  magnification  and  the  rotation  are  constants  for  the  various 
values  of  z  and  hence  the  derivative  itself  is  constant,  say  equal  to  a. 

Writing 

D,w  —  a, 
we  have  upon  integrating, 

w  =  otz  -\-  ^, 

where  /3  is  an  arbitrary  constant  of  integration.  This  constant  of 
integration  represents  a  translation  in  the  plane,  and  by  its  proper 
selection  the  points  of  the  one  configuration  finally  go  over  into  the 
corresponding  points  of  the  similar  configuration,  with  which  the 
desired  conclusion  is  established. 

38.   The  transformation  w  =  ->     If  in  the  general  linear  fractional 

transformation,  we  put  a  =  5  =  0,  /3  =  7  =  1,  we  have  a  very  im- 
portant special  case,  namely, 

1 

w  =  — 

z 

We  shall  now  consider  some  of  the  properties  of  this  transformation. 
If  we  write  z  in  terms  of  polar  coordinates,  we  have 

z  =  p(cos  6  -\-  i  sin  6). 
Hence,  we  may  write 


1                  1 

z      p(cos  0  +  1  sin  d) 

=  i|cos(-fl)  +ism(-e) 
P 

Putting 

w  =  p'(cos  6'  -\-i  sin  6'), 

we  have 

p 

Geometrically,  we  may  consider  this  transformation  as  made  up 
of  two  parts.  Let  the  point  P  (Fig.  68)  represent  any  complex 
number  z.    Draw  through  P  the  line  OP  passing  also  through  the 

origin.     Upon  OP  find  a  point  P'  so  that  OP'  =  p'  =  -•    The  loca- 

P 

tion  of  this  point  may  be  then  considered  as  the  first  step  in  the 
geometrical  interpretation  of  the  given  transformation.  This  oper- 
ation is  called  geometric  inversion.    The  second  step  consists  in 


Akt.  38.] 


THE  TRANSFORMATION  W  =  Z' 


167 


rotating  the  point  P'  about  the  axis  of  reals  until  it  again  falls  into 
the  plane  of  the  paper,  that  is  through  an  angle  of  180°.  We  shall 
call  this  process  a  reflection  upon  the  axis  of  reals.  The  given  trans- 
formation may  be  called  a  reciprocation  and  consists  of  a  geometrical 
inversion  followed  by  a  reflection 
upon  the  axis  of  reals. 

We  shall  first  consider  the  prop- 
erties of  geometric  inversion.  This 
process  is  one  that  belongs  to  ordi- 
nary metrical  geometry.  If  we  draw 
about  the  origin  a  circle  of  unit  radius, 
any  point  upon  this  circle  will  invert 
into  itself,  that  is  it  remains  invariant 
by  the  process  of  inversion.  Every 
point  within  this  unit  circle  is  con- 
verted by  this  process  into  a  point 
lying  without  it  and  vice  versa.  Every 
line  drawn  through  the  origin  is  converted  into  itself,  except  that  the 
points  are  rearranged  upon  the  line.  The  points  very  near  to  the 
origin  are  converted  into  points  lying  at  a  great  distance  and  con- 
versely. As- we  have  already  seen,  it  is  convenient  to  regard  the 
complex  plane  as  closed  at  infinity,  that  is,  as  having  a  single  point 
at  infinity.  This  point  at  infinity  inverts  into  the  origin  and  vice 
versa. 

To  determine  the  character  of  the  configurations  into  which  con- 
figurations other  than  straight  lines  through  the  origin  are  inverted, 
we  shall  now  turn  to  the  analji^ic  side  of  inversion.  Suppose  that  by 
inversion  z  is  changed  into  z'.     Since  this  process  differs  from  the 

transformation  w  =  ~  in  that  the  reflection  upon  the  axis  of  reals  is 

omitted,  we  then  have 

z'  =  -  (cos  0  +  *  sin  ^) 
P 

p(cos  5  +  *  sin  0) 


^  x-\-iy 

X2  +  t/2 

Putting  z'  =  x'  -^r  iy' ,  we  have 

x-\-iy 


P" 


x'  +  iy'  = 


\-i 


y 


x^  -\-  y^      x^  -\-  y^        x^  +  2/2 


168  LINEAR  FRACTIONAL  TRANSFORMATIONS         [Chap.  V. 

or  equating  the  real  parts  and  the  imaginary  parts,  we  get 

Solving  these  equations  for  x  and  y,  we  have 

These  are  the  values  of  x  and  y  which,  if  substituted  in  the  equation 
of  a  given  curve,  give  the  equation  of  the  inverse  curve. 

Ex.  I.   Find  the  curve  into  which  a  straight  line  not  passing  through  the  origin 
is  mapped  by  geometric  inversion. 
The  equation  of  the  given  line  is 

^x  +  By  +  C  =  0,         C  ?f  0. 
Substituting  for  x,  y  their  values  from  (2)  we  obtain 

C(x'2  +  j/'2)  +  Ax'  +  By'  =  0. 

This  is  the  equation  of  a  circle  passing  through  the  origin.  The  equation  of  the 
tangent  to  this  circle  at  the  origin  is 

Ax  +  By  =  0, 

which  is  a  line  parallel  to  the  given  line.  Therefore,  a  system  of  parallel  lines  in- 
verts into  a  system  of  circles  having  a  common  tangent  at  the  origin.  For  C  =  0, 
we  have  the  special  case  of  a  line  through  the  origin  already  discussed. 

Ex.  2.  Find  the  curve  into  which  a  circle  not  passing  through  the  origin  is 
changed  by  inversion. 

The  equation  of  the  given  circle  is  of  the  form 

x^  +  y'  +  2gx  +  2fy  +  C  =  0. 

Substituting  the  values  of  x,  y  from  (2),  we  have 

y'^  ,      2gx'      ,      2fy' 


'     r^'2  _1_  „'2\2     '     ^'2  _L  ,,'2  ~  /r'2  _1_  ,,'2  ~  ^  "» 


(x'«  +  y")'  ^  ix'^  +  y'^Y  ^  x'2  +  y'^  ^  x'^  +  y' 

or  C{x'^  +-y"^)  +2gx'  +  2fy'  +  1=0, 

which  is  the  equation  of  a  circle  not  passing  through  the  origin.  In  the  special 
case  where  C  =  0,  we  have  a  straight  line  not  passing  through  the  origin.  If 
we  now  think  of  a  straight  line  as  a  circle  of  infinite  radius,  we  may  then  make 
the  general  statement  that  by  geometric  inversion  every  circle  is  converted  into 
a  circle. 


Akt.  38.] 


THE  TRANSFORMATION  W  =  Z' 


169 


The  angle  at  which  two  curves  cut  each  other  is  preserved  by- 
geometric  inversion,  but  the  direction  of  the  angle  is  reversed;  that 
is,  the  angle  is  measured  in  the  opposite  direction  after  inversion. 
We  shall  first  show  that  this  statement  holds  when  one  of  the  given 
curves  is  a  straight  line  passing 
through  the  center  of  inversion. 
Let  A,A',B,B'  (Fig.  69)  be  two 
sets  of  points  which  are  inverse 
with  respect  to  0.  Lines  through 
A,  A'  and  B,  B'  pass  through  0. 
Moreover,  we  have 

OA  .  OA'  =  OB  •  OB'  =  L 
The  angle  at  0  is  common  to  the 
two  triangles  OAB  and  OA'B', 
and  as  the  sides  of  the  common 
angle  are  proportional,  the  two 
triangles  are  similar.  Conse- 
quently, 

Z.OAB=  ZOB'A'.       (3) 
If  now  we  think  of  the  points 

A  and  B  as  situated  on  some  curve,  the  points  A'  and  B'  will  lie 
upon  the  inverse  curve.  Let  the  point  B  approach  A  as  a  limit. 
Then  the  point  B'  approaches  the  point  A'  along  the  corresponding 
curve.  The  lines  AB  and  A'B'  become  the  tangents  AT  and  A'T' 
to  the  two  curves  at  the  corresponding  points  A  and  A',  respectively. 
In  the  limit,  therefore,  the  angle  OB' A'  becomes  the  angle  vertical  to 
A  A'B'  and  hence  equal  to  it.     From  (3)  we  then  have  in  the  limit 


ZOAT  =:  LAA'T. 
The  line  OA  is  its  own  inverse  since  it  passes  through  the  center  of 
inversion,  and  by  hypothesis  the  curve  A'B'  is  the  inverse  of  the 
curve  AB.  By  inversion  the  angle  that  the  tangent  to  the  curve  AB 
makes  with  OA,  measured  in  a  clockwise  direction,  namely  LA' AT, 
is  changed  into  the  equal  angle  LAA'T'  made  by  the  tangent  to 
the  inverse  curve  A'B'  with  OA,  measured  in  a  counter-clockwise 
direction. 

Suppose  we  now  consider  the  case  of  any  two  curves  intersecting 
in  a  point  A.  The  inverse  curves  will  intersect  in  a  point  A'  which 
is  the  inverse  of  A.  To  extend  the  argument  to  this  case,  draw  a 
straight  line  through  A,  A' .    It  will  pass  through  the  center  of  in- 


170  LINEAR  FRACTIONAL  TRANSFORMATIONS         [Chap.  V. 

version.  Consider  the  angle  made  by  the  tangent  to  each  curve  with 
this  line  at  the  point  of  intersection.  From  the  foregoing  discussion 
this  angle  is  preserved  in  magnitude  but  reversed  in  direction  by  in- 
version. By  combination  of  these  angles  we  have  the  desired  result; 
that  is,  by  geometric  inversion  angles  are  preserved  in  magnitude  but 
reversed  in  direction. 

Inversion  is  therefore  a  conformal  transformation  but  with  a 
reversion  of  any  given  angle.  Reflection  upon  a  straight  line  is 
likewise  a  process  that  involves  a  reversion  of  angles.     As  we  have 

seen,  the  transformation  w  =  -  is  made  up  of  a  geometric  inversion 

and  a  reflection  upon  the  axis  of  reals.  When  we  combine  these  two 
processes  we  have  a  process  in  which  these  two  reversions  annul  each 

other.     Hence,  we  can  say  that  the  transformation  w  =  -  is  conformal 

without  reversion  of  angles. 

We  have  thus  far  confined  our  discussion  to  geometric  inversion 
with  respect  to  the  unit  circle,  because  of  the  fact  that  inversion  with 

respect  to  this  circle  is  involved  in  the  transformation  w  =  -.    This 

restriction,  however,  is  not  essential  to  the  geometry  of  inversion. 
We  may  define  inversion  with  respect  to  a  circle  of  radius  k  by  merely 
replacing  the  above  condition  pp'  =  1  by  the  more  general  one 
pp'  =  k^.  To  show  that  the  same  geometric  properties  hold  for  the 
general  case  suppose  we  think  of  the  whole  plane  as  being  so  expanded 
or  contracted  about  the  origin  that  the  unit  circle  changes  into  the 
required  circle  of  radius  k.  Any  two  corresponding  points  P,  P' 
with  respect  to  the  unit  circle  become  two  corresponding  points  Q, 
Q'  with  respect  to  the  required  circle.  If  p\,  pi  are  the  radii  vectores 
of  the  points  Q,  Q',  then  we  have 

Pi  Pi  =  kp  •  kp' 

=  kW 

=  k\ 
as  was  required. 

Two  corresponding  points  with  respect  to  a  circle  of  inversion  are 
called  conjugate  points.  The  conjugate  of  any  particular  point  with 
respect  to  a  given  circle  may  be  found  geometrically  as  follows.  Draw 
two  tangents  from  the  given  point  A  to  the  given  circle  (Fig.  70). 
Join  the  given  point  with  the  center  0  of  the  circle.  Connect  the 
points  of  tangency  B  and  C.     The  intersection  of  the  chord  BC 


Art.  38.]  THE  TRANSFORMATION  W  =  Z~^  171 

and  the  line  OA  gives  the  required  point  A'.    For,  from  the  figure, 
the  triangle  AOB  is  a  right  triangle,  having  a  right  angle  at  B,  and 

hence  we  have  f  C^  f\0'^r-'  ^  fi^i  0  ^ 

0B^  =  A'0-A6,        f  ^.     AOJ^B  -CTl^'O^' 

showing  the  two  points  A  and  A'  to  be  conjugate  points. 

The  foregoing  construction  holds  when  the  given  point  A  lies 
without  the  circle  of  inversion.  If  the  given  point  lies  within  the 
circle  of  inversion  the  conjugate  point  may  be  found  as  follows. 
Connect  the  given  point  A  with  the  center  0  of  the  circle;  through 
A  draw  a  line  perpendicular  to  AO,  and  at  the  points  where  this 


Fig.  70.  Fig.  71. 

perpendicular  intersects  the  circle  draw  tangents  to  the  circle.  The 
point  of  intersection  of  these  tangents  gives  the  required  point  A'. 
The  proof  is  similar  to  that  in  the  previous  case. 

The  following  theorems  give  additional  important  properties  of 
inversion. 

Theorem  I.  If  a  given  circle  cuts  the  circle  of  inversion  in  two 
points  A  and  B,  then  its  inverse  cuts  the  circle  of  inversion  in  the  same 
two  points. 

.  The  truth  of  this  theorem  is  seen  at  once  from  the  fact  that  the 
points  of  intersection  of  the  given  circle  with  the  circle  of  inversion 
are  points  on  the  circle  of  inversion  and  therefore  necessarily  invert 
into  themselves.  It  does  not  follow,  of  course,  that  the  given  circle 
as  a  whole  inverts  into  itself. 

Theorem  II.  If  a  given  circle  cuts  the  circle  of  inversion  at  a  given 
angle,  then  its  inverse  cuts  the  circle  of  inversion  at  the  same  angle. 


172  LINEAR  FRACTIONAL  TRANSFORMATIONS         [Chap.  V. 

The  theorem  follows  from  the  fact  that  the  magnitude  of  an  angle 
is  preserved  by  the  process  of  inversion,  and  hence  the  angle  at 
which  the  given  circle  cuts  the  circle  of  inversion  remains  unchanged 
in  magnitude.     It  is,  however,  reversed  in  direction. 

Corollary.  //  a  given  circle  cuts  the  circle  of  inversion  at  right 
angles,  then  the  given  circle  is  identical  with  its  inverse  and  any  straight 
line  through  the  center  of  inversion  cuts  the  given  circle  in  two  conjugate 
points. 

If  the  given  circle  cuts  the  circle  of  inversion  at  right  angles,  then 
by  Theorem  II  its  inverse  also  cuts  the  circle  of  inversion  at  the 
same  angle,  and  since  through  two  points  on  a  circle  but  one  orthog- 
onal circle  can  be  drawn,  the  given  circle  must  be  identical  with 
its  inverse,  as  the  theorem  requires.  The  only  change  that  takes 
place  in  the  given  circle  is  that  the  portion  of  the  circle  without  the 
circle  of  inversion  becomes  after  inversion  the  portion  within  the 
circle  of  inversion.  Since  the  given  circle  inverts  into  itself,  it 
follows  that  any  straight  line  passing  through  the  origin  cuts  the 
given  circle  in  conjugate  points. 

Theorem  III.  Given  a  pair  of  conjugate  points  with  respect  to  a 
fixed  circle.  Any  circle  through  these  points  inverts  into  itself  with 
respect  to  the  fixed  circle  and  cuts  that  circle  at  right  angles. 

One  of  the  two  conjugate  points  must  lie  within  and  the  other 
without  the  circle  of  inversion.  Consequently,  the  given  circle  cuts 
the  fixed  circle  and  the  two  points  of  intersection  invert  into  them- 
selves. These  points  of  intersection  and  the  two  given  conjugate 
points  make  together  four  points  that  the  given  circle  and  the  in- 
verted circle  have  in  common.  Hence,  the  two  circles  must  coincide. 
By  Theorem  II  the  given  circle  and  the  inverted  circle  cut  the  circle 
of  inversion  at  the  same  angle  but  reversed  in  direction.  But  as  the 
inverted  circle  is  identical  with  the  given  circle  each  must  then  cut 
the  circle  of  inversion  at  right  angles. 

Theorem  IV.  Given  a  system  of  circles  such  that  each  circle  parses 
through  two  given  points  and  intersects  a  fixed  circle  at  right  angles. 
The  two  given  points  of  intersection  of  the  system  of  circles  are  then  con- 
jugate points  with  respect  to  the  fixed  circle. 

Let  the  circles  of  the  system  be  inverted  with  respect  to  the  given 
fixed  circle  M.  By  the  corollary  to  Theorem  II,  each  circle  of  the 
system  inverts  into  itself.     It  is  sufficient  for  our  purpose  to  con- 


Abt.  39.] 


GENERAL  PROPERTIES 


173 


sider  two  circles  Ci,  C2  of  the  system.  The  point  P  of  intersection 
lies  on  both  Ci,  and  C2.  After  inversion  with  respect  to  M,  the  point 
P  must  go  into  a  point  within  M  which  likewise  lies  upon  both  Ci 
and  C2.  It  must,  therefore,  invert 
into  the  second  point  of  intersec- 
tion of  these  two  circles,  namely 
P'.     Hence  the  theorem. 

Theorem  V.  Given  two  conju- 
gate points  vnth  respect  to  a  given 
circle.  If  the  circle  is  inverted  with 
respect  to  a  fixed  circle,  the  given 
conjugate  points  invert  into  conju- 
gate points  with  respect  to  the  in- 
verted circle.  Fig,  72. 

Let  M  be  the  given  circle  and  P  and  P'  two  conjugate  points  with 
respect  to  it.  Suppose  the  circle  M  inverts  into  the  circle  M'  with 
respect  to  the  fixed  circle  C,  and  the  points  P,  P'  invert  into  Q,  Q\ 

respectively.  It  is  required  to 
show  that  Q,  Q'  are  conjugate 
points  with  respect  to  M'.  Draw 
any  two  circles  through  the  con- 
jugate points  P,  P';  these  circles 
cut  the  given  circle  M  at  right 
angles.  These  angles  are  pre- 
served by  inversion.  Hence,  the 
circles  through  the  given  conju- 
gate points  and  cutting  M  at 
right  angles  invert  into  circles 
cutting  M'  at  right  angles.  Since 
the  inverse  points  of  P,  P', 
namely  the  points  Q,  Q',  must 
lie  at  the  respective  intersections 
of  these  inverted  circles,  it  fol- 
lows from  Theorem  IV  that  the  points  Q,  Q'  are  conjugate  points 
with  respect  to  the  circle  M'.  With  this  our  theorem  is  demon- 
strated. 


Fig.  73. 


39.   General  properties  of  the  transformation  w  = 


We 


Y^  +  8* 

shall  now  consider  the  general  case  of  a  linear  fractional  transfor- 


/^•'VM  > 


174  LINEAR  FRACTIONAL  TRANSFORMATIONS         [Chap.  V. 

mation.    We  impose  the  condition  upon  the  four  constants  a,  jS,  y,  8, 
that 


7  8 


=  a8-  0y9^O.  (1) 


If  this  determinant  were  equal  to  zero,  we  should  have  —  =  -  and  the 
given  relation  between  w  and  z  would  then  reduce  to 

a 

W  =  -) 
7 

and  all  points  in  the  Z-plane  would  correspond  to  the  same  point 

-  in  the  TF-plane.  By  imposing  the  condition  (1),  we  are  able  to 
7 

.  set  aside  this  trivial  case. 

The  general  linear  fractional  relation  may  be  decomposed  into  the 
three  following  special  cases,  namely: 

(1)  z'  =z-^-, 

7 

(2)  ."  =  i. 

/o\  ^y  —  oc8   „    .   a 

(3)  w  = z —  z"  -\ 

7  7 

This  statement  can  be  easily  verified  by  making  the  substitutions 
indicated  and  thus  obtaining  the  general  linear  fractional  relation 
between  w  and  z.  Geometrically,  we  may  then  consider  the  general 
linear  transformation  as  made  up  of  the  following: 

(1)  a  translation, 

(2)  a  geometric  inversion  followed  hy  a  reflection  on  the  axis  of 
reals, 

(3)  a  rotation  and  a  stretching  followed  hy  a  translMion;  or  what 
is  the  same  thing,  a  logarithmic  spiral  motion  about  the  point  left  invari- 
ant hy  the  third  of  the  foregoing  transformations. 

As  we  have  already  considered  each  of  these  operations,  we  can 
now  formulate  some  of  the  general  properties  of  a  linear  fractional 
transformation.     Among  these  properties  are: 

Theorem  I.  Conjugate  points  with  respect  to  a  given  circle  are 
transformed  by  the  general  linear  fractional  transformation  into  conju- 
gate points  with  respect  to  the  transformed  circle. 


^^"~    -"'• 


Art.  39.]  GENERAL  PROPERTIES  175 

We  have  seen  (Theorem  V,  Art.  38)  that  conjugate  points  with 
respect  to  a  given  circle  remain  conjugate  points  by  inversion.  Since 
reflection  upon  the  axis  of  reals  does  not  disturb  the  relative  position 
of  points  of  a  given  configuration  except  to  reverse  the  direction  of 
the  angles,-  we  may  conclude  that  the  theorem  holds  for  the  special 

transformation  w  =  - ;   that  is,  it  holds  for  the  transformation  (2) 

given  above.  It  also  holds  for  the  transformations  (1)  and  (3)  since 
by  both  these  transformations  the  similarity  of  the  configuration  is 
preserved.  As  the  general  Hnear  transformation  is  decomposable 
into  these  three  special  transformations,  for  each  of  which  conjugate 
points  remain  conjugate  points,  the  theorem  follows  as  stated. 

Theorem  II.  Any  given  configuration  is  mapped  conformally,  ivith- 
out  reversion  of  angles,  by  means  of  a  linear  fractional  transformation. 

This  theorem  follows  from  the  fact  that  each  of  the  three  simple 
transformations  into  which  the  general  linear  fractional  transforma- 
tion may  be  decomposed  is  such  that  the  conclusions  stated  in  the 
theorem  hold. 

Since  w  is  holomorphic  for  all  values  of  z  in  the  finite  region  except 

for  z  = ,  this  same  result  may  be  obtained  independently  by  the 

consideration  of  DzW.     We  have 

„  a8  —  ^y 

{yz  +  5)2 
But  by  hypothesis 

ah-  ^y  9^  0. 

Hence,  by  the  theorem  of  Art.  27,  the  desired  result  follows. 

Theorem  III.  By  the  general  linear  fractional  transformation, 
circles  are  converted  into  circles. 

It  is  here  understood  that  a  straight  line  is  to  be  considered  as  a 
circle  of  infinite  radius.  The  truth  of  the  theorem  follows  from  the 
fact  that  it  holds  for  each  of  the  three  special  transformations  into 
which  the  general  linear  transformation  may  be  decomposed. 

Theorem  IV.  The  general  linear  fractional  transformation  leaves 
two  points  in  the  complex  plane  invariant. 

To  establish  this  theorem,  we  proceed  as  follows.     If  any  point 


176  LINEAR  FRACTIONAL  TRANSFORMATIONS         [Chap.  V. 

2  of  the  complex  plane  is  transformed  into  itself  by  means  of  a  linear 
fractional  transformation,  then  we  must  have 

Z  =  — r> 

72  +  5 

that  is 

722  +  (6  -  a)  z  -  /3  =  0.  (1) 

This  equation  is  a  quadratic  and  has  therefore  two  roots,  namely : 

z,  = ^ ,  22  =  ^ (2) 

The  two  points  21,  22  remain  unchanged  by  the  general  linear  frac- 
tional transformation,  since  each  is  transformed  into  itself. 

These  invariant  points  may  be  finite  and  distinct,  finite  and  co- 
incident, one  finite  and  the  other  infinite,  or  finally,  both  may  be 
infinite.  The  analytic  conditions  for  these  various  cases  may  be 
expressed  in  terms  of  the  coefficients  of  (1).  If  the  discriminant 
vanishes,  that  is  if 

(a -5)2 +  4^7  =  0, 

the  two  roots  of  (1),  that  is  the  two  invariant  points,  are  coincident. 
If  in  addition  we  have  7  =  0,  it  will  be  seen  from  (1)  that  both  roots 
of  (1)  become  infinite;  that  is,  both  invariant  points  coincide  at  the 
point  infinity.  If  7  5^  0  the  two  points  21,  22  lie  in  the  finite  region  of 
the  plane. 
If  we  have 

(a  -  5)2  +  4 187  7^  0, 

the  roots  of  (1),  that  is  the  invariant  points,  are  distinct.  If  in 
addition  7  =  0,  one  of  the  roots  of  (1)  becomes  infinite  and  hence 
one  of  the  invariant  points  is  at  infinity.  It  will  be  observed  that 
when  7  =  0  the  linear  fractional  transformation  reduces  to  the 
general  linear  transformation. 

The  general  linear  fractional  transformation  contains  four  con- 
stants; but  as  we  may  divide  both  numerator  and  denominator  by 
one  of  these  without  affecting  the  transformation,  we  have  only 
three  independent  constants.     We  may  state  the  following  theorem. 

Theorem  V.  There  is  always  one  and  only  one  linear  fractional 
transformation  that  transforms  any  three  distinct  points  into  three  given 
distinct  points. 


Akt.  39.] 


GENERAL  PROPERTIES 


177 


Let  Zi,  Z2,  zs  be  the  three  distinct  points  that  are  to  be  transformed 
into  the  given  distinct  points  Wi,  w^,  w^.  We  must  then  have  the 
three  relations 

aZk-\-  & 


Wk  = 


k  =  l,  2,  3; 


that  is, 


yZk  +  8' 
ywi^k  -\-  Bwk  —  aZk  —  fi  =  0. 


(1) 


Wi   Zi    1 

WiZi  Zi    1 

Wi   Zi    \ 

,              A2  = 

t«222    22    1 

Wz   23    1 

W3Z3   Zz    1 

We  have  given  three  Unear  homogeneous  equations  in  the  four 
unknowns  a,  /3,  7,  5.  The  condition  that  these  equations  have  one 
and  only  one  solution  other  than  a  =  j8  =  7  =  5  =  0is  that  the 
matrix  of  the  coefficients,  or  its  equivalent  matrix, 

W\Z\  W\   2i    1 

1^222    W2    22    1  (2) 

Wt^Z    W3    23    1 

shall  be  of  rank  three;  *    that  is,  that  not  all  of  the  determinants 
formed  from  this  matrix  by  dropping  one  column  shall  vanish. 

We  shall  show  that  this  condition  is  satisfied  by  showing  that  the 
two  determinants 


Ai  = 


can  not  vanish  simultaneously.     Expanding  each  determinant  in 
terms  of  the  elements  of  the  first  column,  we  have 

Ai  =  1^1(22  —  23)  -  w;2(2i  —  23)  +  Wzizx  —  22), 

A2  =  l«i2i(22  —  23)   —  1«222(2i  —  23)  +  ^323(21  —  22). 

Multiplying  the  first  of  these  identities  by  Zi  and  subtracting  the 
second  from  that  result  we  get 

2iAi  —  A2  =  —1^2(21  —  22)  (21  —  23)  +  1^3(21  —  22)  (21  —  23) 

=  (wz  -  W2)(Zi  -  22)  (21  —  23).  (3) 

Since  the  points  Zi,  22,  23  and  Wi,  W2,  Wz  are  distinct,  it  follows  that  (3) 
can  not  vanish.     Hence,  we  have 

2iAi  —  A2  5^  0, 

and  consequently  the  two  determinants  Ai,  A2  can  not  vanish  simul- 
taneously. 

*  See  B6cher,  Introduction  to  Higher  Algebra,  Art.  17. 


178  LINEAR  FRACTIONAL  TRANSFORMATIONS        [Chap.  V. 

Since  the  equations  (1)  have  one  and  only  one  solution  other  than 
a  =  j3  =  7  =  5  =  0,  it  follows  that  any  three  ratios  of  these  un- 
.  knpwns  are  uniquely  determined.  Consequently  there  is  one  and 
only  one  transformation  of  the  required  type  which  transforms  the 
three  distinct  points  Zi,  z^,  Zz  into  three  distinct  points  Wi,  w^,  ws. 
Hence  the  theorem. 

Remembering  that  three  distinct  points  definitely  determine  a 
circle  we  may  now  say  that  any  circle  can  be  transformed  into  any 
other  circle,  or  into  itself,  by  means  of  a  Unear  fractional  transfor- 
mation. Since  the  three  points  upon  the  given  circle  can  be  selected 
in  an  infinite  number  of  ways,  it  follows  that  the  required  transfor- 
mation can  be  made  in  an  infinite  number  of  ways. 

If  it  is  desired  to  transform  four  points  into  four  points,  we  must 
have  an  additional  condition  satisfied.  If  we  have  given  any  four 
points  Zi,  22,  23,  Zi  the  ratio 

Zi  —  02    .     2i  —  Zi 
Zz  —  Z2         Zz  —  Zi 

is  called  the  anharmonic  ratio  or  cross-ratio  of  these  four  points. 

The  following  theorem  gives  the  condition  which  must  be  satisfied 
in  order  that  any  four  distinct  points  Zi,  z^,  zz,  Zi  may  be  trans- 
formed into  four  distinct  points  Wi,  w^,  Wz,  Wi  by  a  linear  fractional 
transformation. 

Theorem  VI.  The  necessary  and  sufficient  condition  that  any  four 
distinct  points  of  the  complex  plane  may  be  transformed  by  a  linear 
fractional  transformation  into  any  other  four  distinct  points  of  the  plane 
is  that  the  anharmonic  ratio  of  the  two  sets  of  points  is  the  same. 

Let  the  four  given  points  be  Zi,  22,  Zz,  24  and  let  it  be  required  to 
transform  these  points  into  the  four  distinct  points  Wi,  W2,  wz,  Wi. 
If  the  four  given  z-points  are  transformed  by  a  linear  fractional 
transformation  into  the  four  given  ly-points,  then  we  must  have  the 
four  relations 

^^  =  ^FjA>        A:  =  1,  2,  3,  4,  (4) 

yzk  -f-  0 

or 

yWkZk  +  8wk  —  az^—  /3  =  0. 

The  necessary  and  sufl&cient  condition  that  these  four  equations  have 


Art.  39.] 


GENERAL  PROPERTIES 


179 


a  solution  other  than  a  =  ^  =  y  =  8  =  0is  that  the  determinant  of 
the  coefficients  shall  vanish;  that  is,  that  we  have  * 


=  0. 


Expanding  this  determinant  in  terms  of  the  last  two  columns  by 
Laplace's  development,  we  have 

2l   1      1 22   1 

23    1         124    1 


WiZi 

Wi 

—  2i 

-   1 

1^^121   Wi   2i    1 

WiZ2 

W2 

-  22 

-   1 

W222   ^^2   22    1 

W3Z3 

Wz 

—  23 

-   1 

w;323  Wz  Zz  1 

WiZi 

Wi 

-  24 

-   1 

w;424  Wi  Zi  1 

WzWi  — 


+ 


WiW2  — 


2i    1  23  1 

22  1  24  1 

23  1  2i  1 

24  1  22  1 

Making  use  of  the  identity 

2i  1  23   1    _    2i 

22  1  24  1  23 


WiWi  + 


zi  1 
24  1 


22  1 
24  1 


2i    1 
23    1 


WiWz  + 


22  1 
24  1 


+ 


2i    1 
24    1 


22  1 

23  1 


22     1 

23  1 


22  1 

23  1 

2i  1 

24  1 


=  0, 


WiWz 


WiWi. 


we  may  write  the  foregoing  relation  in  the  form 

(Zl  —  Z2)(Z3  —  24)   {  (WzWi  +  Wim)  —  {WiWi  +  WiWz)  I 

+  (21  —  24)  (22  —  23)  1  (w^wz  +  WiWi)  —  (w2Wi  +  WiWz)  I 

=  —  (21  —  22)  (23  —  Zi){Wi  —  Wi){wz  —  Wi) 
.+  (2i  —  24)  (23  —  Z2){Wi  —  W'i){Wz  —  Wi)  =  0, 

whence  we  get 

2i  —  22  .  2i  —  24  _  Wi  —  Wj  ,  Wi  —  Wi  . 
23  —  22   *  23  —  24         Wz  —  W2'  Wz  —  Wi' 

that  is,  the  anharmonic  ratio  of  the  four  points  21,  22,  23,  24  is  the  same 
as  the  anharmonic  ratio  of  the  four  points  Wi,  w^,  Wz,  Wi.  As  this 
result  presents  the  necessary  and  sufficient  condition  that  the  equa- 
tions (4)  have  a  solution  other  than  a  =  /3  =  7  =  5  =  0,  it  follows 
that  this  result  also  gives  the  necessary  and  sufficient  condition  that 
the  one  set  of  four  points  may  be  mapped  by  a  linear  fractional 
transformation  into  the  other  set.  Consequently,  the  theorem  fol- 
lows as  stated. 

It  may  be  remarked  that  as  a  consequence  of  the  foregoing  theorem 
a  linear  fractional  transformation  has  the  property  that  it  leaves  the 
anharmonic  ratio  of  any  four  points  invariant. 

If  the  order  in  which  the  four  given  points  are  taken  is  changed 

*  See  B6cher,  Introduction  to  Higher  Algebra,  Art.  17,  Theorem  3,  Cor.  2. 


180  LINEAR  FRACTIONAL  TRANSFORMATIONS        [Chap.  V. 

then  the  anhannonic  ratio  may  be  changed.  Of  the  twenty-four 
ways  in  which  four  points  may  be  selected,  only  six  give  distinct 
anhannonic  ratios.  If  we  denote  any  one  of  these  ratios  by  X,  then 
the  six  are  given  by  * 

X  i  1-X  -^  -^  ^^^. 

^'  X'  '  1-X'         X-1'  X 

That  X  is  in  general  a  complex  number  follows  from  the  fact  that 
it  is  defined  as  the  ratio  of  such  numbers.  It  may  therefore  be  repre- 
sented as  a  point  in  the  complex  plane.  Since  any  two  of  these  six 
anhannonic  ratios  are  linearly  related,  the  geometric  interpretation 
of  these  relations  furnishes  an  interesting  exercise  in  the  application 
of  the  principles  developed  in  this  chapter. 

If  X  describes  a  circle  in  the  complex  plane,  then  the  points  repre- 
senting respectively  the  various  ratios  likewise  describe  circles. 
Moreover,  if  X  is  represented  by  points  within  a  given  region  boimded 
by  a  circle,  it  follows  that  the  other  ratios  are  represented  by  points 
within  regions  bounded  by  circles.  It  is  possible  to  so  choose  the 
region  for  X  that  the  entire  complex  plane  shall  be  filled  by  the  regions 
of  the  six  ratios  without  overlapping.  The  relative  position  of  these 
regions  may  be  found  as  follows.  With  x  =  0  and  x  =  1  as  centers 
describe  two  unit  circles  (Fig.  74).  These  circles  intersect  at  the 
points  whose  coordinates  x,  y  satisfy  the  two  equations 

x2  -h  2/2  =  1, 
(x  -  \y  -t-  2/2  =  1. 

By  solving  these  equations,  we  have 


X  = 


_  1 


y  =  ±  §  Vs. 


The  points  of  intersection  are,  therefore,  —u^  and  —u,  where 

-  1  +  i  VS 
2 

is  one  of  the  cube  roots  of  unity.     If  X  takes  the  values  in  the  unshaded 

region  (0,  |,  —  w*).  Fig.  74,  then-  is  confined  to  the  region  found  by 

A 

inverting  this  region  with  respect  to  the  center  0  and  reflecting  the 
result  upon  the  real  axis.     The  numbers  X  and  1  —  X  are  symmetrical 

with  respect  to  the  point  ^-    The  region  for  r r-  may  be  obtained 

^  1  —  A 

*  See  Scott,  Modem  Analytical  Geometry,  p.  37. 


Art.  39.] 


GENERAL  PROPERTIES 


181 


from  that  for  1  —  X  by  inverting  this  region  with  respect  to  the  circle 
about  the  origin  as  a  center  and  reflecting  the  result  upon  the  axis  of 

reals.    From  the  region  for  -  we  may  find  the  region  for  — r — ,   be- 


FiG.  74. 

cause  these  two  numbers  are  symmetric  with  respect  to  |.  In  this 
way  we  find  the  regions  described  by  the  various  complex  numbers 
as  shown  in  the  figure.  If  X  is  represented  by  the  points  of  a  shaded 
region,  then  the  points  representing  the  other  five  anharmonic  ratios 
are  confined  to  the  shaded  regions. 
There  are  two  important  special  cases  of  anharmonic  ratios.     One 

of  these  cases  is  obtained  if  X  has  such  values  that  X  =  r  and  hence 

X  =  ±1.     For  X  =  —  1,  the  four  points  are  said  to  be  harmonic. 

The  six  ratios  are  then  coincident  in  pairs. 

When  X  is  a  complex  number,  as  in  the  present  discussion,  it 

is  possible  for  three  of  the  anharmonic  ratios  to  be  equal.     For  ex- 

/  .  1 

ample,  it  wiH  be  seen  from  the  figure  that  the  three  ratios  X,  ^^  _  ,  > 

— r —  may  become  equal  at  the  common  point, 

A 

,      1  +  iVs 


182                 LINEAR  FRACTIONAL  TRANSFORMATIONS        [Chap.  V. 
The  reciprocals  of  these  values,  that  is  -,  1  —  X,  r t,  then  become 

A  A  1 

equal  at 

1-iVs 


This  equality  leads  to  the  second  special  case  of  anharmonic  ratios: 
for,  putting 

1  X-  1 

we  have 

X2  -  X  +  1  =  0, 
whence 

^       l±iVs 
2 ' 

which  are  the  two  imaginary  cube  roots  of  —1.  When  X  has  either 
of  these  values  the  four  points  are  said  to  be  equianharmonic.* 

If  the  variables  and  constants  involved  in  a  linear  fractional  trans- 
formation are  all  real,  the  property  that  anharmonic  ratios  are  pre- 
serv'ed  is  commonly  spoken  of  as  a  projective  property;  in  fact  this 
property  m^y  be  made  the  basis  of  projective  geometry.  The  rela- 
tion between  anharmonic  ratios  and  linear  fractional  transformation, 
as  estabhshed  in  Theorems  V  and  Vt,  suggests  the  extension  of  projec- 
tive geometry  to  the  field  of  complex  numbers.  In  the  one  case  the 
single  variable  x  takes  the  totality  of  real  values  and  the  ideal  num- 
ber 00,  represented  by  the  points  on  a  straight  line  including  the 
point  at  infinity.  As  a  result,  we  have  the  projective  geometry  of  a 
straight  line.  In  the  other  case  the  single  variable  z  takes  the  total- 
ity of  complex  numbers  and  the  ideal  number  oo .  Since  but  a  single 
variable  is  involved  this  aggregate  is  sometimes  spoken  of  in  pro- 
jective geometry  as  the  complex  line.  This  extension  of  projective 
geometry  to  the  realm  of  complex  numbers  leads  to  the  consideration 
of  the  theory  of  chains,  f  but,  as  no  use  will  be  made  of  this  theory  in 
the  present  volume,  it  will  not  be  considered  here. 

In  this  connection  it  is  also  of  interest  to  point  out  the  general 
relation  between  the  totality  of  linear  fractional  transformations  and 
the  theory  of  groups.     We  have  for  example  the  following  theorem. 

*  For  a  more  extended  discussion  of  these  cases  see  Harkness  and  Morley, 
Treatise  on  the  Theory  of  Functions,  p.  21,  et  seq. 

t  For  a  discussion  of  this  subject,  see  J,  W.  Young,  AnnaU  of  Math.,  Vol.  II, 
pp.  33-48. 


Abt.  39.] 


GENERAL  PROPERTIES 


183 


Theorem  VII.  The  system  of  linear  Jr actional  transformations 
possesses  the  group  property. 

The  statement  contained  in  the  theorem  involves  the  condition 
that  if  a  Unear  fractional  expression  in  one  variable  is  subjected  to  a 
linear  fractional  transformation,  the  resulting  expression  is  a  linear 
fractional  expression.     Given  the  relation 


w  = 


aiz'  +  /3i 


7i   5i 


Suppose  2'  is  associated  with  z  by  the  relation 


^0. 


z  = 


«2g  +  182 
722  +  62  ' 


a2  182 

72    ^2 


5^0. 


The  theorem  requires  that  w  be  expressed  as  a  linear  fractional 
function  of  2,  where  the  determinant  of  the  coefficients  is  also  differ- 
ent from  zero.     We  have 


w  — 


"22  +  132 
72^  +  O2 
a22  +  /32    ,    . 

71— — TT^  +5i 
722  +  62 

^  (aia2  +  /3i72)  2  +  (ai/32  +  /3ig2) 
(7i«2  +  5i72)  2  +  (7ii32  +  8i8i) 

which  is  a  linear  fractional  expression  in  z.     The  determinant  of  the 
coefl&cients  is  different  from  zero;  for,  we  have 


«1«2  +  /3l72       «l/32  +  /3l52 

7ia2  +  5i72     7i)32  +  5i52 


«!    /3l 

7i  81 


<X2    ^2 

72  82 


and  each  determinant  in  the  second  member  of  this  equation  is  differ- 
ent from  zero  by  hypothesis. 

The  system  of  linear  fractional  transformations  possesses  the  other 
characteristic  properties  of  a  group,*  and  the  relation  to  the  theory 
of  groups  is  at  once  established.     If  a,  /3,  7,  5  are  integers  such  that 

a  /3 


7  5 


=  1, 


then  the  transformation 


w  = 


az-\-  ^ 


72  +  3 
See  B6cher,  Introduction  to  Higher  Algebra,  p.  82. 


184  LINEAR  FRACTIONAL  TRANSFORMATIONS       [Chap.  V. 

defines  the  modular  group.*  Many  of  the  properties  of  linear  frac- 
tional transformations  that  have  been  discussed  follow  also  as  applica- 
tions of  group  thcory.f 

40.  Stereographic  projection.  Since  complex  numbers  are  of  the 
form  z  =  X  -\-  iy,  where  x  and  y  may  vary  independently  of  each 
other,  two  degrees  of  freedom  are  necessary  for  the  geometric  element 
used  to  interpret  them  and  the  plane  naturally  suggests  itseK  for 
that  purpose.  Thus  far  we  have  restricted  ourselves  to  this  mode 
of  representation.  There  are  other  ways,  however,  of  representing 
complex  numbers  and  other  surfaces  than  the  plane  have  been  made 
use  of  in  this  connection.  It  is  frequently  convenient  to  employ  the 
sphere  for  this  purpose.  In  order  to  do  so,  it  must  be  possible  to 
establish  in  some  way  a  one-to-one  correspondence  between  the  points 
of  a  plane  and  those  upon  the  sphere. 

The  desired  result  may  be  accomplished  by  assuming  the  complex 
plane  as  before  and  supposing  that  we  have  a  sphere  tangent  to  this 
plane  at  the  origin.  We  shall  refer  to  the  point  of  tangency  as  the 
south  pole  of  the  sphere,  while  the  opposite  pole  will  be  spoken  of  as 
the  north  pole.  If  we  now  take  the  north  pole  0'  as  the  center  of 
projection  we  can  project  in  a  definite  manner  every  point  of  the 
plane  upon  the  sphere.  Thus  in  Fig.  75  the  point  P  in  the  complex 
plane  corresponds  to  the  point  P'  of  the  sphere.  In  this  way  there 
corresponds  to  each  point  of  the  plane  a  definite  point  of  the  sphere, 
and  conversely.  This  method  of  mapping  the  complex  plane  upon 
the  sphere  is  called  stereographic  projection. 

Since  there  is  a  one-to-one  correspondence  between  the  points 
of  the  complex  plane  and  those  of  the  sphere  the  values  of  z  and  of 
^  —  f{^)  r^ay  be  uniquely  represented  upon  the  sphere,  which  we 
shall  refer  to  as  the  complex  sphere.  For  example,  if  z  describes  a 
continuous  curve  in  a  region  of  the  complex  plane  in  which  w  =  f{z) 
is  holomorphic,  then  w  likewise  describes  a  continuous  curve.  The 
projection  of  these  two  curves  upon  the  sphere  gives  the  interpreta- 
tion upon  that  surface  of  the  relation  between  w  and  z. 

The  point  at  infinity  in  the  complex  plane  projects  into  the  north 
pole  of  the  sphere.  Hence,  to  examine  the  nature  of  a  function  for 
values  of  the  variable  in  the  neighborhood  of  the  point  at  infinity,  it 
may  often  be  convenient  to  represent  both  w  and  z  upon  the  complex 
sphere  and  inquire  into  the  behavior  of  ly  as  2  takes  values  in  the 

*  See  Forsyth,  Theory  of  Functions,  2^1  Ed.,  pp.  680,  681. 

t  See  Kowalewski,  Komplexen  Veranderlichen  und  ihre  Funktionen,  pp.  30-59. 


Art.  40.] 


STEREOGRAPHIC  PROJECTION 


185 


neighborhood  of  the  north  pole.  The  same  result,  of  course,  could 
be  obtained  analytically.  Corresponding  to  the  coordinates  x,  y  of 
a  point  in  the  plane,  we  may  determine  the  location  of  a  point  upon 
the  sphere  by  means  of  two  coordinates  6,  <f>,  one  measured  along 


Fig.  75. 

some  standard  meridian  and  the  other  along  the  equator.  Such  a 
system  of  coordinates  is  a  famiUar  one  in  the  location  of  a  point  upon 
the  earth's  surface  by  means  of  its  longitude  and  latitude.  Any 
given  curve  can  be  mapped  from  the  plane  upon  the  sphere  by 
means  of  the  analytic  relation  between  x,  y  and  the  coordinates  of 
the  corresponding  point  on  the  sphere,  and  the  transformed  function 
thus  obtained  can  be  studied  for  values  of  6,  0  in  the  neighbor- 
hood of  the  north  pole.  A  closed  curve  upon  the  sphere  divides  the 
surface  of  the  sphere  into  two  parts.  This  curve  may  be  regarded 
as  the  boundary  of  either  of  these  regions  to  suit  our  convenience. 
It  is  desirable  to  examine  somewhat  more  closely  into  the  effect 
of  stereographic  projection  upon  the  character  of  a  configuration. 
First  of  all,  suppose  we  have  a  pencil  of  rays  passing  through  the 
origin  and  lying  in  the  complex  plane.  Each  of  these  rays  projects 
into  a  great  circle  passing  through  0  and  0'.  They  become  meridians 
upon  the  sphere  and  one  such  meridian  passes  through  each  point 


186  LINEAR  FRACTIONAL  TRANSFORMATIONS        [Chap.  V. 

upon  the  sphere.  As  a  special  case  the  axis  of  reals  projects  into  a 
meridian  of  reals  and  the  axis  of  imaginaries  projects  into  a  meridian 
of  imaginaries  cutting  the  meridian  of  reals  at  right  angles. 

If  we  have  a  system  of  concentric  circles  in  the  plane  having  the 
origin  as  center,  they  constitute  the  orthogonal  system  to  the  pencil 
of  rays  just  mentioned.  These  circles  go  over  into  the  orthogonal 
system  of  circles  on  the  sphere,  namely,  the  parallels  of  latitude.  One 
of  these  circles  projects  into  the  equator  of  the  sphere.  This  circle 
may  be  conveniently  selected  as  the  unit  circle  in  the  plane.  All 
concentric  circles  lying  within  this  unit  circle  will  become  parallels 
of  latitude  in  the  southern  hemisphere  while  those  lying  outside  of 
this  unit  circle  pass  over  into  parallels  of  latitude  in  the  northern 
hemisphere. 

In  order  to  determine  the  character  of  a  configuration  on  the 
sphere  and  its  relation  to  the  corresponding  configuration  in  the 
plane,  we  shall  now  deduce  the  equations  of  transformation  by 
means  of  which  the  cartesian  space  coordinates  of  any  point  upon 
the  sphere  can  be  expressed  in  terms  of  the  cartesian  coordinates  of 
the  corresponding  point  in  the  plane.  Let  ^,  tq,  f  denote  the  co- 
ordinates of  a  point  on  the  sphere.  Let  the  |-axis  and  the  ir)-axis 
coincide  respectively  with  the  axis  of  reals  and  the  axis  of  imagi- 
naries of  the  plane.  Let  the  f-axis  be  perpendicular  to  the  complex 
plane.  Suppose  the  radius  of  the  given  sphere  to  be  ^.  The  equa- 
tion of  the  sphere  is 

r+T)^+(f-^)^  =  i  (1) 

or  ^'  +  T]2-l-r(f-l)  =0.  (2) 

If  we  now  denote  by  x,  y  the  coordinates  of  any  point  P  in  the  plane, 
the  coordinates  ^,  t),  f  of  the  projection  P'  upon  the  sphere  of  the  point 
P  are  readily  found  in  terms  of  x,  y.     From  Fig.  75  we  have 

of  =  x'  +  y\  (3) 

or'  =  Wo^  -f-  of 

=  x2  +  i/2  +  1,  (4) 

DP''  =  e  +  -n\  (5) 

where  DP'  is  drawn  parallel  to  OP.  The  triangles  OPO'  and  DP'O' 
are  similar,  and  consequently  we  have 

OP  ^  DP' 

&p    of' 


Art.  40.]  STEREOGRAPHIC  PROJECTION  187 


By  use  of  (3),  (4)  and  (5),  we  obtain 

x'-hy'  +  l      o^P' 
As  OP'O'  is  a  right  triangle,  we  have 


(6> 


O'P'^  =  DO' .  00'  =  DO' .  1. 


Since  DO'  =  1  —  f ,  we  have 

OT''  =  1  -  f .  (7) 


From  (6)  we  have  then 

x^-\-y^     ^  ^  +  t)g 
x^  +  y^-\-l       1  -  f  ' 
We  have  also 

■t2 


(8) 


DP'"  =  DO'  •  OD, 

from  which  we  have 

DO' 

or                               f  = 
Finally,  we  have 

^2  +  t]2          x^  +  y^ 

1  -  f       x'  +  y'-\-l 

^        X 

fi    y 

or 

(9) 

(10) 

(11) 

By  use' of  (9)  and  (11)  the  equation  (2)  of  the  given  sphere  may  now 
be  written 

1    I   ^'\^2  I       x^  +  y^  -1         ^ 

2  _        x^  -\-  y^  y^       _   (  y  ^  ^  ^ 

°^  "^  ~_{x^-\-y^+iy'x^  +  y^~  (x^  +  y'  +  lS  ' 

whence         -q  =    „  ,   ^^  1   1 ' 

From  (11)  we  get 


.      X  y 


y  x^  +  2/2  +  1      x2  +  t/2  -I-  1 
Hence,  the  general  relations  between  ^,  tq,  f  and  x,  y  are 

t  = ? ^= y ^=  ^'  +  y'  .  (12) 

^      x2  +  ?/2  +  l'        "^      a:2  +  ^2_^l'         i      a;2  4.^2^1      Vi^y 


188 


UNEAR  FRACTIONAL  TRANSFORMATIONS        [Chap.  V. 


From  these  equations  we  obtain 


X  = 


y 


x'  +  y'  =  Y^'  (13) 

(14) 

(15) 
or  A^  +  liv  +  V  (1  -  f)  =  0.  (16) 

This  equation  is  that  of  a  plane  passing  through  the  north  pole  of  the  sphere. 
The  curve  of  intersection  of  the  plane  and  the  sphere  is  therefore  a  circle  passing 


1-r'     ^    i-f     ■"  ■  ^     i-f 

Ex.  I.   Find  the  stereographic  projection  of  a  straight  line. 
The  equation  of  the  given  line  is  of  the  form 

Ax  +  By  +  C  =  0. 
Substituting  from  (13)  the  values  of  x,  y,  we  have 


1-f ' 1-r 

A{  +  B,  +  C  (1  -  f)  =  0. 


Fig.  76. 

through  the  north  pole.  Hence  every  line  in  the  plane  projects  into  a  circle 
upon  the  sphere  passing  through  the  north  pole.  Any  line  parallel,  say,  to  the 
axis  of  imaginaries  (Fig.  76)  goes  over  into  a  circle  through  0'  tangent  to  the 
great  circle  into  which  the  axis  of  imaginaries  projects,  but  lying  wholly  in  one 
of  the  hemispheres  into  which  that  great  circle  divides  the  sphere.  None  of 
these  lines,  however,  other  than  the  one  through  the  origin,  projects  into  a  great 
circle.  A  system  of  straight  lines  parallel  to  the  axis  of  reals  projects  into  a 
system  of  circles  likewise  passing  through  0'  but  perpendicular  to  the  former 
system,  and  all  of  these  circles  on  the  sphere  are  tangent  at  0'  to  the  great  circle 
into  which  the  axis  of  reals  projects. 


Art.  40.]  STEREOGRAPHIC   PROJECTION  189 

Ex.  2.     Discuss  the  stereographic  projection  of  a  circle  in  the  complex  plane 
whose  equation  is 

x2  +  2/2  +  2ffX  +  2/2/  +  c  =  0.  (17) 

Substituting  from  (13)  the  values  of  x,  y,  and  x^  +  y^  we  have 

^     +2g^^  +  2f-^+c  =  0,  (18) 


1-r  ■      "l-f  '     •'  1-f 

or  r  +  2ff^  +  2/, +c(l-f)  =0,  (19) 

or  {l-c)^  +  2gi  +  2fr,+c  =  0.  (20) 

This  equation  is  that  of  a  plane,  and  the  curve  of  intersection  of  this  plane  and 
the  given  sphere  is  a  circle.  We  may  therefore  conclude  that  by  stereographic 
projection  circles  in  the  complex  plane  become  circles  upon  the  sphere.  These 
circles  do  not  in  general  pass  through  the  origin  nor  through  the  north  pole  of  the 
sphere,  as  we  may  see  from  an  examination  of  equation  (20). 

A  general  property  of  stereographic  projection  is  stated  in  the 
following  theorem. 

Theorem.     The  mapping  of  the  sphere  upon  the  complex  plane,  and 
conversely,  by  means  of  stereographic  projection  is  conformal. 

It  has  been  pointed  out  that  the  general  condition  for  conformal 
mapping  is  that  we  have 

ds  =  M-dS, 

where  ds,  dS  are  differential  elements  of  arcs  upon  the  two  surfaces 
concerned.     From  the  calculus  of  real  variables  we  have 

dS^  =  de  +  drf  +  df^  (21) 

where  dS  relates  to  the  sphere,  and  moreover  we  have 

tfe2  =  dx2  +  dy^  (22) 

where  ds  is  taken  in  the  complex  plane. 
From  (13)  we  obtain 

"^"i-r^d-f)^' 

dy  =  A-.+    '*    - 


Hence 


1  -  r    (1  -  r)^ 

,,       de  +  dr^  .   ^{^d^-\-nd-n)d^       (e  +  ri')d^^  .^„. 


From  equation  (2)  we  get 

^  +  r;  =  r(l-r),  (24) 


190  LINEAR  FRACTIONAL  TRANSFORMATIONS        [Chap,  V. 

whence 

2Ud^  +  T3dT))  =df-2rdr.  (25) 

Substituting  these  values  in  (23),  we  obtain 

-  ^ (1  -  r)^     ' 

from  which  we  get 

1-r 

The  definition  of  conformal  mapping  is  therefore  satisfied,  and 
the  ratio  of  magnification  M  in  passing  from  the  sphere  to  the  com- 
plex plane  is  in  this  case  r •     Similarly,  it  may  be  shown  that 

dS  =  ^-j — „  ,   ■■  ds, 

and  hence  the  mapping  from  the  complex  plane  upon  the  sphere  is 

also  conformal,  having  -r-; — .  ,   ^  as  the  ratio  of  magnification.     As 
■^  a;2  +  y2  +  1 

we  might  expect,  this  ratio  of  magnification  becomes  infinite  at  the 
point  r  =  1>  that  is  at  the  north  pole. 

41.  Classification  of  linear  fractional  transformations.  The 
geometrical  interpretation  of  the  linear  fractional  transformations  of 
the  complex  plane  into  itself  may  be  regarded  as  a  problem  in  kine- 
matics. In  the  present  article  we  shall  undertake  to  classify  these 
transformations  of  the  plane  by  means  of  the  corresponding  motions 
of  the  points  of  the  plane.  We  have  already  seen  that  the  linear 
transformation  given  by  an  equation  of  the  form 

W  =  2  +  fl  (1) 

is  a  translation  of  the  points  of  the  complex  plane.  Suppose  the 
lines  of  motion  be  the  system  of  parallel  lines  AB,  Fig.  77.  Let  this 
system  of  straight  lines  be  mapped  by  reciprocation  with  respect  to 
the  origin.  As  such  a  reciprocation  consists  of  geometric  inversion 
with  respect  to  a  unit  circle  about  the  origin  followed  by  reflection 
upon  the  axis  of  reals,  the  resulting  configuration  is  a  system  of  coaxial 
circles  through  the  origin  having  a  common  tangent  at  that  point. 
The  particular  line  Li  of  the  system  AB  of  straight  lines  which  passes 
through  the  origin  maps  into  that  straight  line  Li  through  the  origin 
which  is  the  reflection  of  the  given  line  with  respect  to  the  axis  of 


Art.  41.] 


CLASSIFICATION  OF  TRANSFORMATIONS 


191 


reals.  Those  lines  lying  below  Li  map  into  circles  tangent  to  Li 
at  the_origin  and  lying  above  it.  Likewise  the  lines  of  AB  lying 
above  Li  map  into  circles  tangent  to  Li  at  the  origin  and  lying  below 
Li.  Corresponding  to  a  motion  along  the  lines  AB,  we  have  a  motion 
of  the  points  along  this  system  of  circles  through  the  origin.  The 
corresponding  directions  of  the  motions  in  the  two  cases  are  indicated 
by  the  arrow-heads.  The  orthogonal  lines  CD  map  by  the  same  re- 
ciprocation into  a  system  of  coaxial  circles  through  the  origin  and 
orthogonal  to  the  first  system  of  circles  as  indicated  in  Fig.  77.     The 


Fig.  77. 


motion  of  the  plane  as  here  indicated  is  called  a  parabolic  motion 

about  the  origin. 

That  a  parabolic  motion  about  the  origin  is  a  linear  fractional 
transformation  of  the  plane  may  be  easily  shown.  Let  z  and  w  be 
the  initial  and  final  positions,  respectively,  of  a  point  moving  along 
one  of  these  circles.  Corresponding  to  these  two  points,  we  have 
two  points  z,  w  upon  some  line  of  the  system  AB,  given  by  the  rela- 
tions 


_      1 

z  =  -) 
z 


-       1 


w  = 


w 


(2) 


As  we  have  seen,  the  points  z,  w  are  associated  by  the  relation  given 
in  (1).  Substituting  in  this  equation  the  values  of  z,  w  given  in  (2), 
we  have 

-  =  -  -\-  n, 

W        Z 


192  LINEAR  FRACTIONAL  TRANSFORMATIONS       IChap.  V. 

or  « =  ^,  (3) 

which  is  a  linear  fractional  relation  between  w  and  z. 

The  points  remaining  invariant  by  a  parabolic  motion  about  the 
origin  may  be  readily  found;  for,  by  comparing  (3)  with  the  general 
form  of  the  linear  fractional  transformation,  we  have 

a=l,         /3  =  0,         y  =  n,         8  =  1. 

Hence,  from  Art.  39,  we  have  as  the  invariant  points 

zi  =  02  =  0; 

that  is,  the  two  invariant  points  are  coincident  at  the  origin. 

If  the  paraboUc  motion  takes  place  about  any  point  zo  5^  0,  the 
relation  between  the  initial  and  final  values  of  the  variable,  namely 
between  z  and  w,  is  still  linear.     For  the  translation 

Zi  =  z  —  Zo,         Wi  =  w  —  zo  (4) 

brings  the  origin  to  the  point  Zo,  and  zi,  Wi  are  respectively  the  initial 
and  final  values  represented  by  the  variable  point  with  respect  to  2o. 
Since  the  motion  about  Zo  is  parabolic,  we  have 

Zi 
Wi  = 


1  +  /X2i 

Putting  for  Zi,  wi  their  values  in  (4)  we  have 

Z  —  Zq 
1  +  /x(z  -  Zo)  ' 
g(l  +  fJiZo)  —  \i.Z^ 
AiZ  +  (1  —  /iZo) 


ifj  —  Zo  = 


or  w 


which  is  a  linear  fractional  relation.  The  points  left  invariant  by  a 
parabolic  motion  about  Zo  are  coincident  at  Zo- 

In  case  of  a  translation,  the  invariant  points  are  given  by  (1) 
and  are  coincident  at  infinity.  Consequently,  a  translation  may  be 
regarded  as  a  special  case  of  a  parabolic  motion  where  the  invariant 
points  coincide  at  infinity.  But  as  we  have  seen,  a  translation  is  a 
linear  fractional  transformation.  We  may  then  conclude  that  every 
parabolic  motion  corresponds  to  a  linear  fractional  transformation 
having  coincident  invariant  points. 

We  shall  now  show  that  conversely  every  linear  transformation 
having  two  coincident  invariant  points  is  a  parabolic  motion.     First 


Art.  41.]  CLASSIFICATION  OF  TRANSFORMATIONS  193 

of  all,  suppose  the  two  invariant  points  coincide  at  infinity.  Then  we 
must  have 

(a -5)2 +  4^7  =  0,         7  =  0, 
whence, 

a  =  5. 

The  general  linear  fractional  transformation  then  reduces  to 

w  =  2  +  -, 
a 

which  is  as  we  know  a  translation,  that  is  a  special  case  of  a  parabolic 
motion. 

If  the  two  invariant  points  coincide  at  a  finite  point  zo,  then  by 
translation  the  origin  can  be  moved  to  this  point.  But  a  translation 
does  not  change  the  form  of  the  lines  of  motion.  By  reciprocation 
these  lines  of  motion  through  the  origin  map  into  lines  of  motion 
having  two  coincident  invariant  points  at  z  =  oo.  But  as  we  have 
seen  such  a  motion  is  a  translation,  and  by  definition  the  motion  along 
the  reciprocal  of  these  lines  is  a  parabolic  motion  about  the  origin 
and  consequently  the  original  motion  is  a  parabolic  motion  about  the 
point  2o. 

Another  important  class  of  motions  is  obtained  when  we  apply 
the  reciprocal  substitution  to  the  general  Unear  transformation, 
which  may  be  written 

w  =  vz  -\-  fjL,  (5) 

where  z,  w  are  respectively  the  initial  and  final  positions  of  the  variable 
z-point,  and  v,  n  are  complex  constants. 

By  the  reciprocal  substitution,  the  point  P  maps  into  some  point 
p',  Fig.  78.  The  pencil  of  rays  passing  through  P  maps  into  circles 
through  P'  and  through  the  inverse  of  the  point  at  infinity,  namely 
the  origin  0.  If  the  transformation  is  such  that  the  half-rays  are  the 
lines  of  motion  in  the  one  case,  then  the  corresponding  lines  of  motion 
in  the  other  case  are  the  circles  passing  through  P'  and  0.  The  direc- 
tion of  the  motion  is  indicated  by  the  arrow-heads.  The  resulting 
motion  is  called  a  hjrperbolic  motion  through  the  fixed  points  0 
and  P'. 

If  the  concentric  circles  about  P  are  considered  as  the  lines  of 
motion,  then  in  the  reciprocal  configuration  the  circles  having  their 
centers  on  OP'  are  lines  of  motion  and  the  circles  passing  through  0 


194 


LINEAR  FRACTIONAL  TRANSFORMATIONS        [Chap.  V. 


and  P'  form  the  orthogonal  system.     The  motion  in  this  case  is  called 
an  elliptic  motion. 

By  a  combination  of  rotation  and  stretching,  we  have,  as  we  have 
seen,  a  logarithmic  spiral  motion.     Corresponding  to  the  logarithmic 


Fig.  78. 

motion  about  P  we  have  after  reciprocation  with  respect  to  the  origin 
what  we  shall  call  a  loxodromic  motion  about  the  points  P'  and  0  as 
indicated  in  Fig.  79. 

Since  hyperbolic  and  elliptic  motions  appear  as  special  cases  of 
a  loxodromic  motion,  it  follows  that  all  three  motions  are  linear 


fractional  transformations  if  it  can  be  shown  that  a  loxodromic  motion 
is  such  a  transformation.  To  show  this,  let  as  before  z,  w  be  the  initial 
and  the  final  positions,  respectively,  of  the  variable  point  on  one  of 
the  curves  of  a  loxodromic  motion  about  O  and  P'.    We  have  then 


_      1-1 
z  =  -,       w  =  -, 
z  w 


(6) 


Abt.  41,]  CLASSIFICATION  OF  TRANSFORMATIONS  195 

where  z,  w  are  the  points  on  the  logarithmic  spiral  corresponding  to 
z  and  w.     The  points  z,  w  are  related  to  each  other  by  equation  (5). 
Substituting  for  z,  w  their  values,  we  get 

w      z        ' 
or 

w  =  -^,  (7) 

flZ  -\-v  ^  ^ 

which  is  the  linear  fractional  relation  required. 

We  have,  by  comparison  with  the  general  linear  fractional  trans- 
formation, 

a  =  1,         ^  =  0,         7  =  At,         5  =  1'. 

Hence,  if  i*  ?«^  1,  we  get 

(a  -  5)2  +  4 /St  F^  0,         7  5^0; 

that  is  the  invariant  points  are  finite  and  distinct.     From  equation 

(2),  Art.  39,  it  will  be  seen  that  the  two  invariant  points  are  0  and • 

Instead  of  the  origin  being  taken  as  one  of  the  invariant  points,  any 
point  Pf  may  be  selected.  Th^econd  invariant  point  is  then  obtained 
by  reciprocation  of  the  point  j>t  infinity  with  respect  to  a  unit  circle 
about  the  point  /^  as  a  center.  But  to  remove  the  origin  to  a  given 
point  is  a  translation  of  the  points  of  the  plane,  which  as  we  know  is 
a  linear  substitution.  Hence  it  can  be  shown  by  the  same  method 
as  that  employed  in  the  discussion  of  parabolic  motion  that  a  loxo- 
dromic  motion  about  any  two  fixed  points  of  the  plane  is  a  Unear 
fractional  transformation  having  those  points  as  the  invariant  points. 
If  one  of  the  invariant  points  is  2  =  <» ,  we  have  then  7  =  0  and  the 
transformation  reduces  to 

az  +  ^      a      .  j8 
^  =  -^  =  6-^  +  6' 

which  gives,  as  we  have  seen,  a  logarithmic  spiral  motion.  We  can 
then  regard  a  logarithmic  spiral  motion  as  a  special  case  of  a  loxo- 
dromic  motion,  where  one  of  the  distinct  invariant  points  is  at  infinity. 
Conversely,  every  linear  fractional  transformation  of  the  points  of 
the  complex  plane  such  that  two  distinct  points  are  left  invariant  is 
some  form  of  a  loxodromic  motion.  If  one  of  these  points  is  finite 
while  the  other  is  at  infinity  we  have  a  logarithmic  spiral  motion, 


196  LINEAR  FRACTIONAL  TRANSFORMATIONS        [Chap.  V. 

which  is  as  we  have  seen  a  special  kind  of  loxodromic  motion.  If 
both  invariant  points  are  finite,  then  by  a  translation  one  of  these 
points  can  be  made  the  origin  and  by  reciprocation  with  respect  to 
this  new  origin  the  original  lines  of  motion  are  mapped  into  lines  of 
motion  having  two  distinct  invariant  points,  one  being  at  infinity, 
that  is  into  a  logarithmic  spiral  motion.  Hence,  by  the  definition 
of  a  loxodromic  motion  the  original  motion  is  of  that  type,  since  the 
translation  which  moves  one  of  the  invariant  points  to  the  origin 
does  not  affect  the  form  of  the  lines  of  motion. 

It  has  already  been  pointed  out  that  a  hyperbolic  motion  and  an 
elliptic  motion  are  special  cases  of  a  loxodromic  motion.  The  condi- 
tions under  which  these  special  cases  arise  depend  upon  the  character 
of  V,  as  may  be  seen  from  an  examination  of  (5).     Suppose  we  put 

V  =  p  (cos  <l>  -\-  i  sin  </>). 

Then,  if  p  7»^  0,  0  5*^  0,  the  logarithmic  spiral  motion  maps  into  the 
loxodromic  motion  about  the  finite  invariant  points;  if  p  ?^  1,  ^  =  0, 
we  have  from  (5)  a  pencil  of  rays  and  after  mapping  we  get  a  hyper- 
bolic motion;  if  p  =  1,  <i>  9^  0,  the  resulting  motion  is  elliptic.  It 
will  also  be  observed,  that  in  case  v  =  \,  the  motion  represented  by 
(5)  is  a  translation,  in  which  case  the  invariant  points  become  coin- 
cident at  infinity  and  after  mapping  by  reciprocation  with  respect 
to  some  finite  point  the  resulting  motion  is  the  parabohc  motion  al- 
ready discussed. 

We  are  now  in  a  position  to  classify  the  various  types  of  linear 
fractional  transformations  of  a  plane  according  to  their  kinematic 
properties.  For,  as  a  result  of  the  foregoing  discussion,  it  follows  that 
every  hnear  fractional  transformation  may  be  interpreted  in  terms  of 
some  one  of  the  motions  already  discussed.  As  we  have  seen,  every 
linear  fractional  transformation  leaves  two  points  of  the  plane  in- 
variant. These  points  must  be  either  coincident,  or  distinct.  If 
they  coincide,  we  have  seen  that  the  transformation  is  a  parabolic 
motion,  where  a  translation  appears  as  a  special  case.  If  the  in- 
variant points  are  finite  and  distinct,  we  have  in  general  a  loxodromic 
motion,  which  reduces  to  a  hyperbolic  motion  or  an  elliptic  motion 
according  to  the  particular  value  taken  by  v.  If  the  invariant  points 
are  distinct,  one  being  at  infinity,  then  our  transformation  is  a  loga- 
rithmic spiral  motion,  which  again  reduces  to  an  expansion  or  a 
rotation  according  to  the  particular  values  given  to  i/  as  already 
noted. 


Art.  41.]  EXERCISES  197 

EXERCISES 

d      1.   Given  in   the  Z-plane  the    two  intersecting  curves  x^  -\-  {y  —  2)^  =  4 
and  2x  +  3?/  —  7  =  0.     Map  these  curves  upon  the  TF-plane  by  means  of  the 

relation  w  =  -.     Show  that  the  resulting  curves  intersect  at  the  same  angle  as 

the  given  curves. 

2.  Map  the  orthogonal  systems  of  straight  lines  u  =  c,  v  =  c  from  the 
TF-plane  upon  the  Z-plane;  first,  by  means  of  the  relation  w  =  z^,  second  by  means 

•of  the  relation  w  =  -^.    How  would  the  two  resulting  configurations  be  related 

if  projected  stereographically? 

y?   3.   Given  the  ellipse  -r  +  ^  =  1-     Subject  this  curve  to  the  transformation 

w  =  Sz  +  10  and  find  the  equation  of  the  resulting  curve.     Draw  the  curve. 

What  general  principle  do  the  results  of  this  mapping  illustrate? 

^    4.   By  applying  the  Cauchy-Riemann  differential  equations  to  the  relation 

1 
w  =  -,  show  that  w  is  an  analytic  function. 
z 

0    5.   A  point  z  moves  with  uniform  speed  of  2  cm,  per  sec.  along  a  circle  whose 

center  is  at  2  -|-  3  i  and  whose  radius  is  4  cm.     Determine  the  path,  velocity,  and 

acceleration  of  w,  where  w  =  2z-\-{l+2i). 

6.  Given  the  three  points  I  -\-  2i,  2  -f-  i,  2  +  3  i.  Determine  the  linear 
fractional  transformation  that  will  map  these  points  into  the  following  points: 
2  -|-  2  i,  1  -|-  3  i,  4.     Find  the  invariant  points  of  the  transformation. 

7.  Let  a  circle  passing  through  the  three  points  2  -|-  3  i,  0,  3  +  2  i  be  given. 
Find  the  conjugate  of  the  point  1  +  2  i  with  respect  to  this  circle.  Construct 
the  curve  into  which  the  circle  through  the  points  1+2  i,  2  +  3  i,  3  +  2i  is 
changed  by  reciprocation  with  respect  to  the  circle  of  radius  2  about  the  origin. 

8.  A  given  translation  of  the  complex  plane  is  represented  by  the  equation 
w  =  z  +  fi,  where  /3  =  2  +  3  i.  Show  in  two  ways  how  the  lines  of  motion  in 
the  complex  plane  can  be  changed  into  lines  of  parabolic  motion  upon  the  sphere 
having  the  coincident  invariant  points  at  the  north  pole.  Explain  why  the  two 
methods  give  the  same  result. 

9.  The  speed  at  the  point  2  +  4  i  of  a  moving  point  in  a  motion  of  expansion 
about  the  point  1  +  2  i  is  2  units  per  sec.  At  the  same  instant  what  is  the 
position  and  velocity  of  the  corresponding  point  in  the  hyperbolic  motion  obtained 
through  reciprocation  of  this  motion  of  expansion? 

10.  When  a  transformation  has  the  property  that  on  being  repeated  once  it 
restores  every  point  of  the  plane  to  its  initial  position,  it  is  called  an  involution. 
Determine  the  conditions  that  must  exist  between  the  coefficients  a,  /S,  y,  6,  in 
order  that  the  transformation 

az  +0 

W  =  — r 

yz  +  8 
shall  be  an  involution. 

11.  Show  that  every  involution  is  an  elliptic  transformation. 

12.  Show  how  the  regions  for  the  anharmonic  ratios  of  four  points  may  be 
represented  on  the  complex  plane  in  case  the  four  points  are  harmonic;  in  case 
they  are  equianharmonic 


CHAPTER  VI 
INFINITE   SERIES 

42.  Series  of  complex  terms.  In  this  chapter  we  shall  con- 
sider some  of  the  general  properties  of  infinite  series  whose  terms  are 
complex  numbers.  We  shall  assume  such  knowledge  of  infinite 
series  as  is  usually  given  in  elementary  texts  on  algebra  and  calculus. 
Special  attention  will  be  directed  to  the  properties  of  power  series. 

Suppose  we  have  given  the  series 

Sa„  =  «]  +  a2  +  a3_+   •  •  •   +  "n  +   •  '  '  ,  (1) 

where  an  =  an  ■+■  ibn,  an,  hn,  being  real  numbers.  As  in  series  of 
real  terms,  we  shall  denote  the  sum  of  the  first  n  terms  by  *Sn,  that 
is,  we  shall  put 

Sn  =  ai  -\-  a2  +   •  •  •  +  an. 

A  series  is  said  to  be  convergent  if  the  sequence 

Si,  S-i,    .     .     .    ,  Sn,    .    .    . 

has  a  limit.     If  a  is  this  limit,  that  is  if 

L   Sn  =  a, 

n—co 

then  the  series  is  said  to  converge  to  the  number  a.  This  number 
is  also  called  the  sum  of  the  series  and  is  uniquely  deterrmned  by  the 
series.    We  may  therefore  write 

2a„  =  a. 

If  the  limit  of  Sn  does  not  exist  as  n  increases  indefinitely,  then  we 
say  that  the  given  series  is  divergent. 

The  geometric  significance  of  convergence  may  be  easily  seen. 
All  that  is  needed  is  to  locate  in  the  complex  plane  the  points  that 
represent 

Si  =  OCi, 

Si  =  ai-\-  at, 

S3  =  ai -\-  Oi  -{-  as, 

Sn  =  ai -\-  a2 -{-  as  -\-   •  •  •   +  an, 
198 


Art.  42.] 


SERIES  OF  COMPLEX  TERMS 


199 


It  will  be  noticed  that  these  points  do  not  necessarily  lie  along  a 
straight  line  as  in  series  of  real  terms,  but  that  they  may  be  distrib- 
uted over  the  complex  plane. 
They  are,  however,  so  located 
as  to  be  dense  at  the  limiting 
point  a. 

In  order  that  the  limit  shall  exist, 
it  is  necessary  that   L   |  a:„  |  =  0. 


As  in  series  of  real  terms,  this 
condition  is  a  necessary  but  not 
a  sufl&cient  condition  for  the  con- 
vergence of  the  given  series. 

As  already  stated,  each  term 
of  a  series  of  complex  numbers 
may  be  written  in  the  form 


Y 

h 

s, 

1 
1 

I 

St 
Ss 

s 

r 

sj 

■ji 

n~ 

Sl\ 

$i 

0 

a 

X 

Fig.  80. 


Separating  the  real  and  the  imaginary  parts  of  the  terms  of  the 
series,  we  may  put 

A„  =  ai  +  oo  +  •  •  •  +  a„, 
5„  =  6i  +  62  +  •  •  •  +  &„. 

We  may  now  formulate  the  following  theorem. 

Theorem  I.     The  necessary  and  sufficient  condition  thai  the  series  of 
complex  numbers 

Oil -\-  a2 -\-  as  +   •  •  •   +  a„  +   •  •  • 


converges  to  a  limit  a  =  a  -\-  ib  is  that 

L  An  =  a,  L  Bn 

n=oo  n=oo 

We  have 


(2) 


and  by  Theorem  I,  Art.  12,  the  necessary  and  sufficient  condition 
that  the  sequence 


01,  02,  03,  . 
has  a  limit  a  =  a -\-  ib  is  that 

L  An  =  a, 

n=oo 

Hence  the  theorem. 


,  Sn 


n=oo 


200  INFINITE  SERIES  [Chap.  VI. 

The  point  a  Is  determined  upon  the  axis  of  reals  (Fig.  80)  by  the 
series  2a„,  while  b  is  determined  upon  the  axis  of  imaginaries  by  the 
series  26„.     The  limiting  point  a  is  then  identical  with  a  +  ib. 

It  will  be  seen  at  once,  therefore,  that  if  either  of  the  series  Han, 
Hbn  is  divergent,  the  series 

2a„  =  2(a„  +  ibn) 
is  divergent. 

Ex.1.     Giventhe8eries2(^  +  ii)=(l  +^^)+(i  +  |i)  +  (|  +  ^i)+  •  •  •  • 
This  series  is  divergent;  for,  we  have 

X'^n  =   l+l  +  l  +  l+    ■    ■    ■    , 

where  Zon  is  divergent  although  S6n  is  convergent. 

We  may  also  state  the  necessary  and  sufficient  condition  for  the 
convergence  of  a  series  of  complex  terms  as  follows: 

Theorem  II.     The  necessary  and  sufficient  condition  that  the  series 
"1  +  "2  +   •  •  •   +  a„  +   •  •  • 

converges  is  that  for  an  arbitrarily  small  positive  number  c  there  exists 
a  definite  number  m  such  that 

I  ocn+i  +  an+2  "   *   *   +  ocn+p  |  <  c,      n  >  m,     /)  =  1,   2,  3,   .   .   .   . 

This  condition  follows  at  once  from  Theorem  VI,  Art.  12.  We 
have  the  sequence 

Si,  O2,  S3,    '    •    '    ,    Sn,    .... 

The  necessary  and  sufficient  condition  for  the  convergence  of  this 
sequence  is  that,  for  an  arbitrarily  small  6,  a  corresponding  integer 
m  may  be  found  such  that 

I  Sn+p  —  Sn\  <  €,     n>  m,    p  =  1,  2,  3,  .  .  .  . 

Replacing  the  values  of  /S„  and  Sn+p  by  their  values  in  terms  of  the 
a's,  we  have  at  once  the  condition  stated  in  the  theorem. 

Theorem  III.  The  series  Han  converges,  provided  the  series  of 
moduli  of  the  various  terms  of  the  given  series  converges. 


Akt.  42.]  SERIES  OF  COMPLEX  TERMS  201 

We  have  by  hypothesis  the  condition  that  the  series  of  absolute 
values 

|ai|  +  |a2|+   •  •  •  +|«n|+  •  •  • 

converges.     From  Theorem  II  it  follows  that  since  the  foregoing 
series  of  moduli  converges,  we  have 

{  I  a»+i  I  +  •  •  •  +  I  ocn+p  I  1  <  e,         n>  m,         p  =  1,  2,  3,  •  •  •  . 
But  we  have 

I  Oin+l  +     •    •    •     +  OCn+p  1    —    1  an+l  I  +     *    *    '     +   1  "n+p  1, 

whence  it  follows  that 

I  «„+i  +  •  •  •  +  an+p  I  <  e,        n  >  m,        p  =  1>  2,  3,  •  •  •  . 

By  Theorem  II  this  inequaUty  gives  a  sufl&cient  condition  for  the 
convergence  of  the  given  series  IIa„. 

Ex.  2.     Let  it  be  required  to  test  the  convergence  of  the  series  2an,  where 

p''(cos  nd  +  i  sin  nO) 

a„  = 

n 

The  series  of  moduU  is 

which  converges  f or  p  <  1 ;  for,  we  have  by  the  ratio  test 

„=oo  \n  +  1'   n\        „J„  '' '  n  +  1       ''         '     ^^  7     CI 

Hence,  by  Theorem  III  the  given  series  converges  f or  |  ««  i    <  1. 

If  the  series  of  absolute  values  of  the  various  terms  of  the  series 
Sa„,  namely  2  |  «„  |  =  S  Va„2  +  b„^,  converges,  then  the  given  series 
2a„  is  said  to  converge  absolutely. 

Theorem  IV.  Given  the  series  IIa„,  where  an  =  an  -\-  ihn.  The 
necessary  and  sufficient  condition  that  this  series  converges  absolutely 
is  that  the  two  series  Son,  26„  converge  absolutely. 

.  That  the   absolute  convergence  of  Han  and  II6„  is  a  necessary 
condition  may  be  shown  as  follows.     We  have 

\an\  =ia„  +  i6„|=  Va„2  +  6„2,     I6„|  ^  |an  +  i6„|=  V^iJ+K\     (3) 


We  assume  that  the  given  series  converges  absolutely;   that  is,  that 
the  series  of  moduH  converges.     We  have  then  the  convergent  series 

2  I  a„  I  =  Vai2  +  6i2  +  •  •  .  +  Va„2  +  6„2  +  •  •  •  .  (4) 


202  INFINITE  SERIES  [Chap.  VI. 

From  (3)  it  follows  at  once  that  the  terms  of  the  two  series  2  |  a„  |, 
S  I  6n  I  are  equal  to  or  less  than  the  corresponding  terms  of  the 
convergent  series  (4)  and  hence  both  of  the  series  Hon,  26„  con- 
verge absolutely. 

The  condition  set  forth  in  the  theorem  is  also  sufficient.    If  we 
write 

a;  =  I  ai  I  +  I  02 1  +  •  •  •  +  I  a„  I, 

5;  =  I  6i  I  +  I  62 1  +  •  •  •  +  I  6n  I, 
we  have 

that  is,  since  21  |  an  |,  2  |  6„  |  converge,  the  series  2  { 1  a„  |  +  ]  6„  [  | 
also  converges.     We  have,  however, 

I  an  I  =  I  a„  I  +  I  6n  I, 

and  consequently  the  series  2  |  a„  |  converges;  that  is,  the  given 
series  converges  absolutely. 

St" 
—  for  absolute  convergence. 

We  may  write  the  given  series  in  the  form 


from  which  we  have 


Each  of  these  series  is  convergent,  but  neither  converges  absolutely.     Hence  the 
given  series  can  not  converge  absolutely.     In  fact  we  have 

which  is  a  well-known  divergent  series. 

While  absolute  convergence,  as  we  have  seen,  gives  a  sufficient 
condition  for  the  convergence  of  2a„,  it  is  not  a  necessary  condition. 
It  may  occur,  as  we  have  just  seen  (Ex.  3),  that  a  series  converges 
even  if  the  series  of  moduli  is  divergent. 

v^  z" 
Ex.  4.     Given  the  series    7 ,  — ,  where  z  =  cos  6  +  isind,  Q  <  d  <2ir. 
*^  n 

It  will  be  observed  that  the  series  in  Ex.  3  is  a  special  case  of  the  series  here 


Art.  42,]  SERIES  OF  COMPLEX  TERMS  203 

considered,  namely,  where  ^  =  x  •  The  series  of  moduli  is  2^  ~  >  which  is  a 
divergent  series.     However,  the  given  series 

2a„  =  ll{an  +  ibn) 
is  convergent,  since  each  of  the  two  series 

XV  1          /,            -   ,  cos2«  ,  CO83  0  , 
On  =  2/  n  ^^^       =  cos  0  H 2 ^'  —3 H  •  •  •  , 

X,         V  1    •      zj        .     .    ,  sin20  ,  sinSd   , 
6„  =  2^  -  sm  n0  =  sm  (?  H 1 ^ H  •  •  • 

is  convergent  for  0  <  0  <  2  tt.* 

Absolutely  convergent  series  have  certain  properties  not  pos- 
sessed by  series  in  general.  Some  of  these  properties  are  stated  in 
the  following  theorems. 

Theorem  V.  The  sum  of  an  absolutely  convergent  series  of  complex 
terms  is  independent  of  the  order  of  the  terms. 

When  we  have  a  finite  series  the  order  of  arrangement  of  the 
terms  is  a  matter  of  indifference.  The  foregoing  theorem  enables 
us  to  extend  this  commutative  property  to  an  infinite  series,  pro- 
vided it  converges  absolutely. 

Let  the  given  series  be 

2a!„  =  ari  +  Qr2  +   •  •  '   +  "n  +   •  •  •  ,  (5) 

where  ««  =  fln  +  ibn- 

By  Theorem  IV  the  two  series  2o„,  26n  converge  absolutely. 
When  a  series  of  real  terms  converges  absolutely  the  sum  of  the 
series  is  not  dependent  upon  the  order  in  which  the  terms  of  the 
series  are  arranged.f    Since  we  have 

Sttn  =  2(a„  +  ibn)  =  2a„  +  iS6„  (6) 

and  the  terms  of  the  two  series  2a„,  S6„  can  be  taken  in  any  order 
without  affecting  their  sum,  it  follows  from  (6)  that  the  sum'  of  the 
given  series  Han  is  likewise  independent  of  the  order  in  which  its 
terms  are  taken. 

The  associative  law  of  addition  may  be  extended  to  any  convergent 
infinite  series;  that  is,  we  can  always  insert  parentheses  at  pleasure 
in  an  infinite  series.     However,  if  the  series  converges  absolutely  the 

*  See  Bromwich,  Theory  of  Infinite  Series,  p.  159. 
t  Ihid.,  p.  64. 


204  INFINITE   SERIES  [Chap.  VI. 

sum  is  not  affected  by  any  arbitrary  rearrangement  and  grouping  of 
the  terms;  that  is,  we  have  the  following  theorem. 

Theorem  VI.  The  sum  of  an  absolutely  convergent  series  remains 
unchanged  if  the  terms  are  rearranged  and  grouped  in  any  arbitrary 
manner. 

Let  2a„  be  any  absolutely  convergent  series  haying  the  sum  a. 
Let  us  first  suppose  the  terms  of  this  series  to  be  grouped  by  putting 
into  each  group  a  certain  number  of  consecutive  terms.  Let  Ai, 
Ai,  .  .  .  denote  these  groups,  respectively,  where* 

Ai  =  ai  -\-   •  •  •   -\-  ak, 

At  =  OCk+l  +    •    •    •    +  ttr, 


We  shall  now  establish  the  convergence  of  the  series 

^1  +  ^2  +  •  •  •  +  A„  +  .  •  .  .  (7) 

The  sum  of  the  first  n  terms  of  this  series,  namely, 

5„'   =  ^1  +     •    •    •     +  An, 

is  equal  to  the  sum  of  the  first  m  terms  (m  >  n)  of  the  given  series 
2a„.  As  n  becomes  infinite,  m  also  increases  without  limit.  We 
have  for  all  values  of  n 

Sn     =  Sm,  (8) 

where  Sm  denotes  the  sum  of  the  first  m  terms  of  the  given  series 
Uttn.  However,  m  does  not  take  all  integral  values,  but  increases 
through  the  values  k,  r,  s,  .  .  .  .  Since  the  given  series  converges, 
it  follows  by  Art.  12  that  the  limiting  value  of  Sm  is  the  sum  of  the 
given  series,  namely  a.  As  (8)  holds  for  all  values  of  n,  it  follows 
that  we  have  also 

L  Sn'  =  a. 

Since  the  given  series  converges  absolutely  the  sum  of  the  series 
is  not  affected  by  any  arbitrary  rearrangement  of  its  terms.  Conse- 
quently, in  order  to  get  any  desired  grouping  of  the  terms  of  the  given 
series  2a„,  all  that  is  necessary  is  to  so  rearrange  the  terms  of  the 
series  that  the  terms  occur  in  the  required  order  and  then  form  the 
series  (7)  of  groups  of  consecutive  terms.     Hence  the  theorem. 

Convergent  series  that  do  not  converge  absolutely  are  called 
conditionally  convergent  series.  The  following  theorem  may  be 
stated  with  reference  to  such  series,  namely:  « 


Art.  42.]  SERIES  OF  COMPLEX  TERMS  205 

Theorem  VII.  The  sum  of  a  conditionally  convergent  series  de- 
pends upon  the  order  of  arrangement  of  its  terms. 

Let  Han  be  the  given  conditionally  convergent  series.  As  in  the 
demonstration  of  Theorem  V,  we  have 

2a„  =  Il(a„  +  ibn)  =  I!a„  +  iI16„. 

By  Theorem  IV  we  know  that  at  least  one  of  the  series  Han,  26„ 
must  converge  conditionally;  otherwise  the  given  series  would  con- 
verge absolutely.  Suppose  that  II6„  alone  converges  conditionally. 
It  follows  from  (6)  that  the  given  series  Sa„  converges  to  a  limit 
a  +  ix,  where  rr  =  2I6„  depends  upon  the  arrangement  of  the  terms 
of  the  series  Hbn,  and  by  properly  choosing  this  arrangement  x  may  be 
made  any  arbitrarily  chosen  real  number.*  This  establishes  the 
theorem. 

In  this  demonstration  we  have  made  use  of  the  fact  that  in  a 
conditionally  convergent  series  of  real  terms  the  series  may  be  made 
to  converge  to  any  arbitrarily  chosen  real  number.  This  fact  would 
seem  to  suggest  that  a  conditionally  convergent  series  of  complex 
terms  might  likewise  be  made  to  converge  to  any  arbitrarily  chosen 
complex  number  as  a  limit.  This,  however,  is  not  in  general  the 
case.  As  already  pointed  out,  if  116„  converges  conditionally,  we 
may  so  arrange  the  terms  of  this  series  as  to  make  it  converge  to  any 
arbitrarily  chosen  number  x.  In  order  that  the  series  2a„,  by  a 
rearrangement  of  its  terms,  shall  approach  an  arbitrarily  chosen 
complex  number  x  +  iy  it  is,  in  general,  also  necessary  that  IIa„  be 
conditionally  convergent  and  that  the  rearrangement  of  its  terms  be 
unrestricted.  This,  however,  is  impossible,  for  when  the  6's  are 
properly  arranged  the  a's  must  be  taken  in  such  an  order  that  there 
is  associated  with  each  hk  an  a^  such  that  ajt  +  ihk  is  one  of  the  as. 
Since  the  choice  of  the  arrangement  of  the  a's  is  thus  restricted,  it 
follows  that  while  the  limit  of  a  conditionally  convergent  series  of 
complex  terms  depends  upon  the  order  of  its  terms,  it  can  not,  in 
general,  be  made  to  approach  any  complex  number  at  pleasure  by  a 
suitable  arrangement  of  its  terms.  Some  particular  conditionally 
convergent  series  of  complex  terms  may,  however,  be  made  to 
approach  any  previously  assigned  limit  by  the  proper  arrangement 
of  the  terms. t 

*  See  Bromwich,  Theory  of  Infinite  Series,  p.  68. 
t  See  Levy,  Nouv.  Ann.  Math.,  Vol.  5,  1905,  p.  506. 


206  INFINITE   SERIES  [Chap.  VI. 

To  test  the  convergence  of  a  series  of  complex  terms,  we  need  only 
to  employ  the  methods  of  testing  series  of  real  terms.  We  can  always 
make  the  convergence  of  any  series  2a!„  depend  upon  the  conver- 
gence of  the  two  series  of  real  terms  2Ia„,  2&„,  where  a„  =  a„  +  ihn, 
by  use  of  the  Theorem  I.  Frequently,  it  is  more  convenient  to 
test  the  convergence  of  the  given  series  by  considering  the  series 
of  moduli.  As  we  have  seen,  if  this  series  converges  then  the  given 
series  converges.  The  series  of  moduli  is  a  series  of  real  positive 
terms  and  the  tests  for  the  convergence  of  such  series  may  be  at 
once  applied.  If  the  series  of  complex  terms  converges  condition- 
ally, we  have  no  general  tests;  but  this  is  to  be  expected,  since  no 
general  tests  exist  for  the  conditional  convergence  of  series  whose 
terms  are  real. 

In  the  discussion  thus  far  we  have  considered  only  those  series 
whose  terms  are  constants.  The  terms  of  the  series  may,  however, 
be  functions  of  a  complex  variable.     Such  a  series  may  be  written 

ui{z)  +  ih(.z)  +  •  •  •  +  Un{z)  +  •  •  •  . 

The  convergence  of  this  series  for  a  given  value  of  the  variable  im- 
plies that  if  this  value  is  substituted  for  the  variable,  the  resulting 
series  of  constants  converges.  The  region  of  convergence  may  or 
may  not  be  a  closed  region.  A  series  of  functions  of  a  complex 
variable  defines  a  function  of  the  complex  variable  in  the  region  of 
convergence. 

43.  Operations  with  series.  In  order  to  make  use  of  series  in 
mathematical  discussions,  it  is  necessary  to  establish  the  conditions 
under  which  convergent  infinite  series  may  be  combined  by  the 
fundamental  operations  of  arithmetic  following  the  formal  laws 
applicable  to  sums  of  a  finite  number  of  terms.  It  follows  at  once 
from  the  laws  governing  operations  with  limits  and  from  the  defini- 
tion of  convergence,  that  any  convergent  series  may  be  multiplied 
or  divided  termwise  by  a  constant,  and  that  a  constant  may  be  added 
to  or  subtracted  from  any  convergent  infinite  series  by  adding  it  to  or 
subtracting  it  from  any  term  of  this  series,  precisely  as  in  the  case  of 
a  sum  of  a  finite  number  of  terms.  We  shall  now  consider  the  sum, 
difference,  product,  and  quotient  of  two  convergent  infinite  series. 

Theorem  I.  If  2a„,  ll/3„  are  two  convergent  series  having  the 
limits  a,  /3,  respectively,  then 


Art.  43.]  OPERATIONS  WITH  SERIES  207 

//  the  given  series  converge  absolutely,  then  the  series  (1)  converges 
absolutely. 

Let  Sn  =  ai  -^  a2  -\-   '-•-{-  ant 

T^n  =  ^1  +  )32  +     •    •    •     +  /3n. 

Then,  we  have 

5n  +  r„  =  (ai  +  ^i)  +  (a2  +  /Sa)  +   .  .  .  +  (a„+^„).  (2) 

By  hypothesis  we  have 

L   Sn  =  a,        L    T,  =  ^; 

hence,  it  follows  that 

whence 

a  +  ^  =  (ai  +  /3i)  +  (a2  +  ^2)  +   •  •  •   +  («„  +  /3„)  +   •  •  •  . 

We  shall  now  show  that  if  2]a„,  2/3„  converge  absolutely,  the 
series  (1)  converges  absolutely.     Let 

I  "n  I    =  Pn,        I  /3n  I    =  /"n,       ^p„   =  p,       2r„   =  T, 
'Sn     =  Pi  +  P2  +     •    •    •     -\-  Pn, 

T'n'  =  r:  +  r2  +  •  •  •  +  r„. 

Putting  Pn  =  Pn-{-  r„, 

we  may  write 

Pn  =  Pl  +  P2  +     '    '    •    +  Pn. 

We  have  then 

L   P.=    L   (Sn'  +  no  =  p  +  r; 

that  is, 

P  +  r  =  (pi  +  ri)  +  (p2  +  ro)  +  ...  +  (p„  +  r„)  +  •  •  •  .      (3) 

Since  we  have 

I  "n  +  /3„  I  =  I  a„  I  +  I  /3„  I  =  p„  +  r„, 

it  follows  from  (3)  that  the  series  of  moduli  of  (2)  converges,  that  is 
(2)  converges  absolutely. 

It  is  to  be  noted  that  the  parentheses  in  (1)  may  be  removed; 
for  the  sum  of  the  first  n  terms  of  the  series 

ai  +  /3i  +  "2  +  /32  +  as  +  •  •  • 

differs  from  the  sum  of  a  properly  chosen  number  of  terms  of  (1) 
by  at  most  the  first  term  in  one  of  the  parentheses,  this  term  ap- 
proaching zero  with  increasing  n. 


208  INFINITE  SERIES  [Chap.  VI. 

Ex.  1.    Add  the  two  series 

log(l+z)  =z-W  +  hz'-\z'  +  ^z'-  •  ■  •  , 

log  (1  -  Z)   =    -Z  -  §  Z2  -  §  Z3  -  1  2^  -  ^  Z5  -    .    .    .    . 

Applying  the  results  of  Theorem  I,  we  obtain 

log  (1  -  z2)  =  log  (!+«)+  log  (1  -  z) 

The  corresponding  theorem  for  the  difference  of  two  series  may  be 
stated  as  follows: 

Theorem  II.  If  2a„,  HPn  are  two  convergent  series  having  the 
limits  a  and  /3  respectively,  then 

a-i3=2(an-j8„)  =  (ai-/3i)  +  (a2-^2)+  •  •  •  +(an-^n)+  •  •  •  •   (4) 

If  the  given  series  converge  absolutely,  then  the  series  (4)  converges 
absolutely. 

The  first  part  of  this  theorem  may  be  established  by  a  method 
similar  to  that  employed  in  the  demonstration  of  Theorem  I  and 
the  demonstration  need  not  be  repeated.  To  show  that  series  (4) 
converges  absolutely,  we  have 

I  an  -  ^n  I    =    1  an  I   +    I  /3n  I    =   Pn  +  r„.  (5) 

In  the  demonstration  of  Theorem  I,  it  was  shown  that  the  series 
2(pn  +  r„)  converges.  Hence  from  (5)  it  follows  that  the  series 
2  I  a„  —  /3„  I  must  converge;  that  is  (4)  converges  absolutely. 

Ex.  3.    From  the  series 

log  (1+2)   =Z-U'  +  W-lz*-\-lz^-    •    •    • 

subtract  the  series 

log(l  -  z)  =  -z  -  W"  -  W  -  Iz"  -  }i^  -  •  •  •  . 
By  Theorem  II  we  get 


log 


(r^)  =log(l+z)-log(l-z)  =2z  +  |z»+?z«  + 


The  following  theorem  due  to  Cauchy  *  gives  a  convenient  crite- 
rion for  the  multiplication  of  series. 

Theorem  III.  If  the  two  series  2a„,  2I/3„  converge  absolutely  to  the 
limits  a  and  /3,  respectively,  and  if  each  term  of  the  one  series  is  multi- 
plied into  each  term  of  the  other,  the  series  whose  terms  are  these  pro- 
ducts taken  in  any  order  which  includes  them  all  in  a  simple  infinite 
series  converges  absolutely  and  has  the  limit  a^. 

*  Enc.  der  Math.  Wiss.,  lA,,  p.  96;  Enc.  des  Sci.  Mat.,  h,  p.  247. 


Art.  43.]  OPERATIONS  WITH  SERIES  209 

If  each  term  of  ^an  is  multiplied  into  each  term  of  I!j8„  the  result 
may  be  schematically  displayed  as  follows: 

tti  /3i  +  ai';/32  +  ort'iiSs  +    •  •   •   +  ai/3„  +    •  • 

.«^'A±«2-i'^2  +a2\pi+   •  •  '  +a20n  +•  • 

-,-'.'.'_'_-.«i-.^L'+.«3A +.«3li33  +   •  •  •  +  a3,5„:.+  •  •  •  I  (g) 

a„  /3,  +  a„  /32  +a„  03  +   .   .  .    +  a„0„  +  •  • 

The  theorem  announces  that  if  we  make  up  a  series  by  selecting  the 
terms  of  this  scheme  in  any  manner  so  that  each  term  appears  in 
the  resulting  series,  then  that  series  converges  absolutely  and  has 
the  limit  a0. 

Suppose  we  arrange  the  terms  of  the  foregoing  array  into  a  series 
in  the  following  order: 

«i|8i  +  ai/32  +  a202  +  0:2181  +  aijSs  +    •  •  •  .  (7) 

The  order  of  selection  of  the  terms  of  the  series  follows  the 
boundaries  of  successive  squares  from  top  to  bottom  and  then  from 
right  to  left  as  indicated  in  (6).  We  shall  now  show  that  series 
(7)  converges  absolutely.     Put  as  before 

I  "n  I    =  Pn,  1  jSn  1   =  ^n, 

and  let  the  series  Sp„,  Sr™  converge  respectively  to  the  limits  p 
and  r.    We  may  write  for  convenience 

Sn     =  Pi  +  />2  +     •    •    •     +  Pn, 

T„'  =  ri  +  r2  +  •  •  •  +  Tn. 

It  is  proposed  to  show  that  the  series  of  moduli  obtained  from  (7), 

namely 

Piri  +  Pir2  +  P2r2  +  P2ri  +  pirs  +  •  •  •  ,  (8) 

converges.     Let  Mk  denote  the  sum  of  the  first  k  terms  of  this  series; 

then  if  we  have 

n2  S  fc  =  (n  +  1)2, 
it  follows  that 

Sn'Tn     —   Mk   =  S  n+lT  n+l- 

But  since  both  2a„,  2j3„  converge  absolutely  it  follows  that 
L  Sn'Tr,'  =  L  SUiTUi  =  L  Sn'  L  Tn'  =  pr. 


210  INFINITE  SERIES  [Chap.  VI. 

Consequently,  we  have,*  since  k  increases  with  n, 

L  Mk  =  pr 

ifc=oo 

and  series  (7)  converges  absolutely. 

Since  the  series  (7)  converges  absolutely  its  terms  may  be  grouped 
in  any  manner  desired  without  affecting  the  limiting  value  of  the 
series.     Suppose  we  collect  the  terms  of  (7)  into  groups  as  follows: 

ai/3i  +  (ai/32  +  a2^2  +  «2^i)  +   •  •  • 

+  (aiiSn  +   •  •  •   +  a„^„  +   •  •  •  +  a„^i)  +   •  •  •  •  (9) 


+  an/3i) 


For 

convenience  put 

Vn 

=  (a  A  +   •  •  • 

+  an^n-\-     • 

and 

Pn  = 

Pi  +  P2  + 

•    •    •    +Pn, 

Sn  = 

ai  +  "2  + 

•    •    •    +  «n. 

Tn  = 

/3i  +  /32  + 

•  •  •  +  /3„. 

We  have  then 

Pn   = 

=  ^nTn- 

Upon  passing  to  the  limit  as  n  becomes  infinite,  we  obtain 

L    P„   =      L     SnTn  =  L    Sn    L    Tn   =  ajS. 


0  =  00       n=oo 


Since  the  limit  of  the  series  (9)  is  equal  to  that  of  (7),  it  follows  that 
the  latter  series  not  only  converges  absolutely  but  has  the  limiting 
value  a^. 

The  series  (7)  was  obtained  by  selecting  its  terms  from  (6)  in  a 
particular  manner.  It  has  been  shown,  however,  that  (7)  converges 
absolutely  and  consequently  by  Theorem  VI,  Art.  42,  its  terms  may 
be  rearranged  and  grouped  in  any  manner  desired.  It  follows  then 
that  whatever  is  the  manner  of  arranging  the  terms  of  (6)  into  a 
series  the  limit  is  still  a/3  and  the  resulting  series  converges  abso- 
lutely. It  is  often  convenient  to  select  the  terms  of  our  series  by 
taking  the  terms  of  (6)  along  diagonal  lines  as  follows: 

aii3i+(au82  +  a2/3i)-|-   •  •  •+ (aij8„+a2|8n-i  +  •  •  •  +  an^i)  +  •  •  •  (10) 

In  this  arrangement  of  the  product,  each  group  contains  all  of  the 
terms  a^iS.  for  which  r  +  s  has  the  same  value.  From  the  foregoing 
discussion,  it  follows  that  this  series  likewise  converges  absolutely 
and  has  the  limit  a/3. 

*  See  Townsend  and  Goodenough,  First  Course  in  Calculus,  p,  24,  Theorem  VI. 


Art.  43.]                          OPERATIONS  WITH  SERIES 
Ex.  3.    Given  the  two  series, 

Z3            ^            ^ 
sin2=2-3-,  +  g-,-;^+    ..    .    , 

^2            2^            2« 
C0S2  =  l-^  +  ^-^+    •   •   •, 

.u-l 

to  show  that  2  sin  z  cos  z  =  sin  2  z. 

We  have 

(         z^        z^        z' 
2sinzcosz  =  2J2- -,  +  57-77  +  •  •  • 

"^2!'    2!3!  *  2J51 

+  ^'           ''     4-  .  .  .    • 

^4!      4!3  !^ 

-a^+-- 

_L 

-  .  .  . 

=  2z      8^V^^^      ••• 
"^"^       3!.+  5! 

_                (2z)3       (2z)'> 
^^^^         3!     '     5! 

=  sin  2  z. 

Mertens  has  shown  *  that  the  series  (10)  converges  to  the  limit  afi 
when  only  one  of  the  given  series  2a„,  S/3„  converges  absolutely. 
It  may  be  shown  f  that  when  both  of  the  series  2a„,  2/3„  converge 
conditionally  then  if  the  series  (10)  converges  at  all,  it  has  the 
limiting  value  ajS.  In  neither  of  these  cases,  however,  is  the  product 
necessarily  an  absolutely  convergent  series.  On  the  other  hand,  the 
series  (10)  may  still  be  absolutely  convergent  in  particular  cases  when 
one  of  the  series  2a„,  2I/3„  is  absolutely  convergent  and  the  other 
conditionally  convergent,  or  even  divergent;  also  when  one  of  these 
series  is  conditionally  convergent  and  the  other  divergent;  and 
finally,  when  both  are  conditionally  convergent  or  both  divergent. | 

As  division  is  the  inverse  operation  of  multiphcation,  the  results 
of  Theorem  III  may  be  used  in  determining  the  conditions  under 
which  we  may  divide  one  series  by  another.  The  resulting  theorem 
may  be  stated  as  follows: 

*  See  Jour,  fur  d.  reine  u.  angew.  Math.,  Vol.  79,  p.  182;  also  Whittaker, 
Modern  Analysis,  Art.  19. 

t  See  Whittaker,  Modem  Analysis,  Art.  20. 

X  See  Cajori,  BuUetin  Amer.  Maih.  Soc,  Jan.  1903. 


212  INFINITE  SERIES  [Chap.  VI. 

Theorem  IV.    Given  the  series  2 «„  and  2 j8n  having  the  limits  a  and  jS, 
respectively.    Suppose  2/3„  converges  absolutely  and  that  /3i  9^  0.     Then 

^  =  ||-"  =  Xl  +  X2  +    •    •    •    +  Xn  +    .    •    .    ,  (11) 

,  ..  Ctn  —  Xx/3„  —  X2j8n-1  —     •    •    •     —  \n-lP2 

where  X„  = > 

Pi 

provided  the  series  2X„  converges  absolutely. 
In  order  that  we  may  have  the  relation 

the  series  San  must  be  the  product  of  2j8„  and  SXn.  Since  both  of 
the  series  2j8„,  SX„  converge  absolutely,  they  may  be  multiplied 
term  by  term  and  the  product  written  in  the  form 

XA+(Xii32+X2^i)+  •  •  •  +(Xi|8„+X2i8„-i+  •  •  •  +X„-i/32+X„^i)+  •  '  •  • 

If  this  series  is  to  be  identical  with  the  series 

ai  +  "2  +•••+«„+•••  , 

then  the  value  of  Xi,  X2,  .  .  .  ,  X„,  .  .  .  must  be  so  determined  that 
the  corresponding  terms  of  the  two  series  are  equal;  that  is,  we  must 
have 

ai  =  XiiSi, 

OC2  =  \].^2  +  X2/3i, 


an  =  Xii8„  +  X2i3n-i  +  •  •  •  +  X„_i/32  +  X„/3i, 
whence  we  have,  since  /3i  9^  0, 


Xi 

X2 

«2 

-Xxft 

^1    ' 

^      _  «n   —   Xi/3n   —  X2i3n-1  —     •    •    •     —  Xn-1^2 
An ^  : 


This  completes  the  demonstration  of  the  theorem. 


Art.  44.]  DOUBLE  SERIES 

Ex.  4.     Given  the  two  series 

gS  g5  g7 

z^       z*       ^ 


find 


t&nz  =  Z  +  —  +  —  + 


We  have,  since  /3i  =  1, 
Xi  =  ai  =  2, 
^  ^  «  23  /       2M       2»> 

X2  =  «2  -  Xi^i  =  -  3-!-2^-2lj  =3^ 

Xa  =  a,  -  Xi^3  -  X202  =  5-,  -  2  (^^  j  -  -(^-  ^j  =  — , 


2»> 


213 


44.   Double  series.     Let  us  consider  a  doubly  infinite  array  of 
complex  elements  of  the  form 


(1) 


an 

an 

an    . 

■    OCln      .    .     . 

CX21 

0.13, 

a23    . 

•    a2n      ... 

asi 

a32 

ass    . 

.  asn    ... 

CXml 

am2 

OimZ    . 

.     .    Olmn    .     .     . 

where  «„„  indicates  the  element  in  the  w"*  row  and  the  n""  colunm. 
Each  row  of  the  array  extends  indefinitely  to  the  right  and  each 
column  extends  indefinitely  downwards.  If  the  elements  of  this 
array  are  connected  by  plus  signs,  the  result  is  called  a  double  series, 
denoted  by 

2amn.  (2) 

Let  Smn  denote  the  sum  of  the  elements  in  the  first  m  rows  and  n 
columns  of  the  given  array.  If  Smn  approaches  some  definite  limit 
a  as  w  and  n  become  infinite  independently  of  each  other  and  if  this 
limit  is  independent  of  the  manner  in  which  m  and  n  become  infinite, 
that  is  if 


L     Smn   =  OC, 


(3) 


n=QO 


then  the  double  series  (2)  is  said  to  converge  to  the  limit  a,  and  a  is 
called  the  sum  of  the  series.  If  the  limit  (3)  does  not  exist,  the 
series  is  said  to  be  divergent.     If  the  series  formed  by  taking  the 


214  INFINITE  SERIES  .  [Chap.  VI. 

moduli  of  the  elements  of  (1)  converges,  then  the  given  series  is  said 
to  converge  absolutely. 

The  product,  of  two  series  2a„  •  Si3„,  already  discussed,  may  be 
exhibited  as  a  double  series.  We  need  merely  connect  the  various 
rows  of  (6)  in  the  previous  article  with  the  plus  sign.  The  double 
series  was  in  that  case  converted  into  a  simple  series  by  adding 
the  elements  of  the  array  along  the  two  sides  of  successive  squares. 
The  order  of  the  terms  in  the  resulting  simple  series  is  that  of  (7), 
Art.  43.  If  this  series  converges  absolutely,  then  by  Theorem  V, 
Art.  42,  it  converges  independently  of  the  order  of  the  terms.  The 
terms  can  therefore  be  so  rearranged  as  to  give  various  simple  series 
equivalent  to  the  given  double  series,  every  such  series  converging 
to  the  same  limit,  namely  the  limit  of  the  double  series.  We  may 
say,  moreover,  that  if  the  moduli  of  the  elements  of  the  double  series 
Sctmn  can  be  arranged  in  any  one  way  so  as  to  form  a  simple  conver- 
gent series,  then  the  double  series  Sa^n  converges  absolutely.  Such 
a  double  series  may  then  be  converted  into  a  simple  series  in  any 
arbitrary  manner  by  means  of  which  there  is  set  up  a  one-to-one 
correspondence  between  the  totality  of  elements  of  the  double  series 
and  the  positive  integers  representing  their  order  as  terms  in  the 
simple  series. 

The  following  generalization  of  Theorem  VI,  Art.  42,  makes  the  use 
of  absolutely  convergent  double  series  of  advantage  in  some  dis- 
cussions. 

Theorem.  //  a  double  series  of  complex  terms  converges  absolutely, 
it  may  he  evaluated  by  rows  or  columns. 

Let  Hctmn  bp  the  given  double  series  whose  terms  are  the  elements 
in  the  array  (1).  Suppose  this  series  converges  absolutely  and  has 
the  limit  a.  We  are  to  show  that  the  limits  of  the  series  constitut- 
ing the  various  rows  exist  and  that  the  sum  of  these  limits  forms  a 
series  Having  a  as  a  limit.  Furthermore,  we  are  to  show  that  the 
limits  of  the  series  constituting  the  various  columns  exist  and  that 
the  sum  of  these  limits  likewise  forms  a  series  having  the  limit  a  of 
the  double  series. 

Let  the  series  of  moduli  of  the  terms  of  Hamn,  namely  2Ip„„,  con- 
verge to  the  limit  p.  The  series  formed  by  taking  the  moduli  of  the 
elements  of  any  row  must  converge;  for,  the  sum  of  a  finite  number 
of  such  terms  increases  with  n  and  is  always  less  than  p.  We  shall 
denote  the  limiting  values  of  the  successive  rows  of  moduU  by  pi, 


Art.  44.1  DOUBLE  SERIES  215 

P2,  .  .  .  ,  Pm,  •  •  .  .  Since  each  row  of  the  series  of  moduli  con- 
verges it  follows  from  Theorem  III,  Art.  42,  that  each  row  of  the 
series  Hamn  converges.  Let  the  successive  rows  converge  respectively 
to  the  limits  on,  0:2,  as,  ...  ,  am,  •  •  .  ,  and  let  Sm  denote  the  sum  of 
the  first  m  terms  of  this  series. 

We  shaU  now  show  that  the  series  Som  converges  to  the  limit  a. 
This  series  will  converge  if  the  series  of  moduU  2  |  a^  |  converges. 
We  have  for  all  values  of  m 

\oCm\    —  Pm, 

and  since  the  sum 

Pi  +  P2  +     •    •    •     +  Pm 

increases  with  m  and  is  always  less  than  p,  the  series  Spm  converges, 
and  by  comparison  21  |  «„  |  converges.  Therefore  Sa^  converges, 
say  to  a'. 

It  remains  to  show  that  a  =  a.  Denote  by  Ri,  R2,  .  .  .  ,  Rm,  re- 
spectively, the  absolute  values  of  the  remainders  after  the  first  n 
terms  of  the  first  m  rows  of  the  given  double  series  Samn)  that  is,  let 

Ri  =  I  at,  n+i  +  a.-,  n+2  +  •  •  •   I,    i  =  1,  2,  •  •  •  ,    n>  m. 

For  an  arbitrarily  chosen  c,  we  may  now  select  a  number  Nm  de- 
pending upon  w,  such  that  for  n  >  Nm  each  of  the  numbers  Ri, 

R2,  .  .  .  ,  /2m  is  less  than  —  •    We  have  then 

m 

\Smn-Sm\=Rl  +  R2+     ■    •    '    +Rm<e,      U  >  Nm.  (4) 

However,  since  the  series  Hom  converges  to  the  limit  a',  we  have  for 

some  value  mi, 

I  ,S,„  -  a'  I  <  €,         m  >  wi.  '  (5) 

Combining  (4)  and  (5),  we  obtain 

I  aS„„  -  a'  I  <  2  €,        m  >  mi,        n>  Nm.  (6) 

Since  the  Hmit    L  Smn  exists,  we  have  from  (6) 


m=oo 
n=oo 


L    kimn   — 
m  =00 
n=oo 


By  hypothesis  Smn  has  the  limit  a  as  m  and  n  become  infinite.  Hence, 
a'  must  be  identical  with  a  and  the  series  Ho™  converges  to  a  as  the 
theorem  requires. 


216  INFINITE  SERIES  [Chap.  VI. 

A  similar  argument  shows  that  when  the  given  double  series  2a„„ 
is  evaluated  by  columns,  the  limiting  value  is  again  a,  the  sum  of 
the  given  series. 

Ex.  Consider  the  following  double  series,  which  is  of  imp>ortance  in  the  Weier- 
Btrassian  theory  of  elliptic  functions: 

...    I  i  I  1  I       ^        I  ^  I  ^  I 

^  (-4«i+4«,)»  ^  (-2«i+4w3)^  ^  (4(03)»  ^  (2«,+4w,)3  ^  (4wi+4w3)3  ^ 

,  1  I  1  111 

'  '  '   ~^  (-4«i+2a)s)»  "^  (-2«i+2w3)»  "^  (2«3)»  "^  (2wi+2«s)''  '^  (4coi+2w3)'  "^ 

'■'"•"  (-4a,i)»       "^  (-2a,i)»       "^  ^      "^  (2i:;o '      "^  (4i:^«      "*" 

■  1  ,  1  111 


(-4wi-2«,)»^(-2a>i-2w3)'  '  (-2«3)'     (2wi-2w,)»  '  (4wi-2w,)» 
(-4«i-4«,)»^(-2«i-4«3)''^(-4«3)''^(2coi-4coj)»^(4wi-4«3)'' 


where  wi  and  ws  are  any  two  complex  numbers  subject  only  to  the  restnction  that 
the  real  part  of  ( -  —  )  shall  be  greater  than  zero.     The  points 

ii  =  2  TOiwi  +  2  msws, 

where  mi,  mz  are  integers,  Ue  on  a  network  of  parallel  lines  covering  the  entire 
complex  plane. 

The  series  under  consideration  is 

where  S'  denotes  the  sum  for  all  values  of  Q  except  the  value  for  which  OTi,  ms 
are  both  equal  to  zero. 

This  double  series  may  be  converted  into  a  simple  series  and  its  absolute  con- 
vergence established,  by  selecting  the  points  Q  in  order  along  the  sides  of  the 
successive  parallelograms  as  indicated  in  Fig.  81. 

Let  these  successive  parallelograms  be  denoted  by 

Pi,  Pit  Pi,    •    •    •   I   Pnj    .    •    ■    . 

Let  I  be  the  least  and  L  the  greatest  distance  of  any  point  of  Pi  from  the  origin. 
On  the  perimeter  of  Pi  are  8  points  Q,  such  that  for  each  point  i  ^ 

II  =£l 

a»|~z»*  I     ' 

On  the  perimeter  of  Pn  are  8  n  points  Q,  such  that  for  each  point 


Abt.  45-] 
Hence,  we  have 


UNIFORM  CONVERGENCE 


217 


n  =1 


Since  the  terms  of  the  given  series  are  less  in  absolute  value  than  the  correspond- 
ing terms  of  the  well-known  convergent  series  included  in  the  braces  multiplied 

by  a  constant  ~,  it  follows  that  the  given  double  series  converges  absolutely. 


-6M,+e<<j; 


JT^IS7  6«,+6ctfj 


i^X 


—6ii>i~6ux 


Fig.  81. 
It  may  be  remarked  that,  on  the  other  hand,  the  double  series 

does  not  converge;  for,  on  the  perimeter  of  Pn  are  8n  points  i2,  such  that  for 
each  point  we  have 


{uLY 


Hence,  we  obtain 


n=l 


{nh)''      L2  r  ^  2  ^  3  ^ 


!• 


where  the  series  inclosed  in  the  braces  is  known  to  be  divergent. 


45.   Uniform  convergence.     Suppose  we  have  given  a  series  of 
functions 

Uxiz)  +  'Wtiz)  +    •    •    •    +  Wn(2)  +    •    •    •    , 

and  suppose  this  series  converges  for  all  values  of  z  in  a  given  region 
<S  which  may  or  may  not  be  closed.     We  shall  speak  of  &  as  the 


218  INFINITE  SERIES  [Chap.  VI. 

region  of  convergence.  In  this  region  the  series  then  defines  a 
function,  and  we  may  write 

m  =  L  S„(3), 

n 

where  Sn{z)  =  ^  Uniz).   We  may  also  write 
1 

f{z)    =  Sn{z)  +  Rn{z), 

where  Rn{z)  represents  the  remainder  after  the  first  n  terms. 

It  is  often  true  that  the  given  series  will  converge  more  rapidly 
in  the  neighborhood  of  certain  points  than  in  the  neighborhood  of 
others.  Let  Zi  be  some  point  in  ;S.  Since  the  series  converges  for 
this  value  of  z,  it  is  possible  to  find,  for  an  arbitrarily  small  positive 
number  e,  a  positive  integer,  say  mi,  such  that  for  all  values  of  n  >  mi, 
we  have 

\f{Zi)-SM)\^\Rn{Zx)\<^. 

If  the  value  of  e  is  kept  constant,  it  will  in  general  be  necessary  to 
select  a  new  integer  W2  if  zi  be  replaced  by  some  other  value  22  of  S. 
If,  in  the  selection  of  m,  the  least  integer  that  will  answer  the  pur- 
pose is  taken,  then  with  each  point  z  there  is  associated  a  particular 
integer  m,  namely  the  first  integer  for  which  we  have  |  Rn{z)  \  <  e, 
where  n>m.  We  may  then  write  m{z)  as  a  function  of  z.  Consider 
now  the  totality  of  all  the  values  of  m  corresponding  to  the  points  of 
the  region  S.  These  values  of  m  may  or  may  not  have  a  finite  upper 
limit.  In  case  a  finite  upper  limit  exists,  we  may  say  that  the  series 
converges  uniformly  in  the  region  *S.  Denoting  the  upper  limit  of 
m{z)  by  M,  then  for  a  given  e  any  integer  m  >  M  may  be  associated 
equally  well  with  each  value  of  z.  The  definition  of  uniform  con- 
vergence may  now  be  stated  as  follows: 

The  given  series  is  said  to  converge  uniformly  in  a  given  region  S, 
closed  or  not,  if  corresponding  to  an  arbitrarily  small  positive  number  e  it 
is  possible  to  find  an  integer  m,  which  is  independent  of  z,  such  that 
for  all  values  of  n  >  m  we  have  simultaneously  for  all  values  of  z,  in 
the  region  S, 

\f(z)-Sn(z)\^\Rn(z)\<e. 

The  following  example  furnishes  an  illustration  of  regions  of  uni- 
form and  of  non-uniform  convergence  of  a  series  of  functions. 


Art.  45.J 

Ex.  4.     Given  the  series 

2" 


UNIFORM  CONVERGENCE 


219 


Z^  +  T 


+ 


+ 


+ 


+ 


1  +Z2    '     (1  +Z2)2     '  '     (1  +z2)n-l 

This  is  a  geometric  series;  hence,  we  have 

We  shall  consider  the  character  of  the  convergence  of  this  series  for  finite  values 
of  2  =  p(cos  d  -\-  isva.6)  in  the  region  defined  by  the  inequalities 


P^O, 


l  =  '-l- 


In  this  region  the  series  converges  and  defines  the  fimction 


7 


/(2)  =  1+2S 
=  0, 


for    2^0, 
for    2  =  0. 


While  the  given  series  converges  in  the  region  indicated,  that  region  is  not  to  be 
understood  as  the  whole  of  the  region  of  convergence.  The  remainder  after  the 
first  n  terms  is,  in  the  region  under  consideration. 


Rn{z)   = 


1 


(1  +22)»-l' 

=  0,  for    2 


for    2  ?f  0, 

.0.  ^ 


For  an  arbitrarily  small  e,  there  corresponds  to  each  value  of  2  an  integer  m  such 
that 

I  Rn{z)  I  <  «,         n>  m. 

But  there  is  no  integer  m,  however  large  it  may 
be  chosen,  that  answers  this  purpose  simultaneously 
for  all  values  of  2;  for,  suppose  we  take  m  =  G, 
chosen  as  large  as  we  please,  then  for  ni  >  (7  we  may 
always  find  values  of  2  such  that 


I  Rn,{z)  I   ^ 


1 


(1  +  Z^)ni-^ 


>  e. 


We  need  only  choose  2  so  that  i  z  1  is  suflBciently 
small.  It  follows  then  that  the  given  series  is  non- 
uniformly  convergent  in  the  region  selected. 

Suppose  we  now  restrict  the  region  by  excluding 
a  region  about  the  origin;  that  is,  let 

p=  r,    where  0  <  r  <  1. 


Fig.  82. 


The  lower  limit  of  |  1  +  2*  I  in  the  new  region  is  then  Vl  -\-  r*,  which  is  the  value 
of  I  1  +  2*  I  when  2  is  at  P  or  Q,  Fig.  82.     Hence,  the  upper  limit  of 


I  Rn(z)  I    - 


1  I 

(1  +  2*)"-!  I 


220  INFINITE  SERIES  [Chap.  VI. 

for  a  given  n  is  —^ .     In  order  to  determine  a  value  of  m  such  that  for 

(1  +  r«)"2" 
«very  z  of  the  region  we  have 

I  Rniz)  I  <  €,        n>m, 

(l+r*)    2 
whence  m'  =  1  — 


log(l+f-) 

Then  for  all  values  of  n  >  m  >  m',  we  have 

1 


\RnZ\^ 


<*» 


(1  +z2)"-i, 
irrespective  of  the  value  of  2  in  the  finite  region  where 

In  this  region  then  the  given  series  converges  uniformly. 

It  will  be  observed  that  the  region  of  uniform  convergence  may  not 
coincide  with  the  region  of  convergence.  In  fact,  it  is  frequently 
more  restricted  than  the  region  of  convergence.  As  a  convenient 
test  for  uniform  convergence,  we  have  the  following  theorem,  due  to 
Weierstrass. 

o/^^^""^    Theorem  I.    Given  the  series 

Ul{z)   +  1*2(2)   +     •     •    •     +  Un(z)   +     •    •    •    . 

If  for  all  values  of  z  in  a  given  region  S,  closed  or  not,  we  Jiave  for  all 
values  of  n 

I  Un{z)   \^Mn, 

where  ^Mn  is  a  convergent  series  of  positive  constants,  then  Huniz) 
converges  absolutely  and  uniformly  in  S. 

The  absolute  convergence  of  Sw„  follows  at  once  from  the  fore- 
going discussion  of  absolute  convergence,  since  by  the  conditions  of 
the  theorem  we  have  |  Un(z)  \  —  Mn  and  2IM„  is  convergent. 

The  uniform  convergence  of  the  series  may  be  established  as 
follows.  Since  the  series  2M„  is  convergent,  we  can  find  a  number 
m  such  that 


Art.  45.]  UNIFORM   CONVERGENCE  221 

Moreover,  we  may  write 

Xun(z)     =  2^"  <  2^"'  p  =  1,  2,  3,   .  .  .  . 

m+1  m+1  m+1 


Consequently,  we  have 

»i+p 


<e. 


Since  this  relation  exists  independently  of  the  value  z  may  take  in 
the  region  S,  the  condition  set  forth  in  the  definition  of  uniform 
convergence  is  satisfied. 

It  is  to  be  noted  that  the  foregoing  theorem  gives  a  sufficient  but 
not  a  necessary  condition  for  uniform  convergence.  Other  and  more 
delicate  tests  for  uniform  convergence  may  be  made  to  apply  to 
series  of  functions  of  a  complex  variable,*  but  the  one  given  is  sufii- 
cient  to  test  those  series  to  be  considered  in  the  present  volume. 

Uniform  convergence  gives  a  useful  criterion  ior  the  continuity 
of  the  function  defined  by  a  convergent  series  of  functions.  This 
criterion  may  be  stated  as  follows : 

Theorem  II.     Given  the  function  f(z)  defined  by  the  convergent  series 

Ui(z)  +  Uiiz)  +    •    •    •    +  Un{z)  +     •    •    •    . 

//  Un(z)  is  continuous  and  the  series  converges  uniformly  in  a  region  S 
closed  or  not,  then  f{z)  is  continuous  in  S. 

We  may  write  the  given  function  in  the  form 

f{z)  =Sniz)i-Kiz). 

Since  the  series  converges  uniformly  in  a  given  region  *S,  it  follows 
that  for  any  value  zq  in  *S  we  have,  for  n  >  m, 

|i2n(20)|<6.  (1) 

Suppose  z  takes  an  increment  AiZ  such  that  Zo  +  AiZ  lies  in  S.  We 
then  have 

I  Rnizo  +  Ai2)  I  <  e, 
whence 

I  ARniZo)   I    =    I  Rn{Zo  +   AlZ)   -  RniZo)   |    <  2  6.  (2) 

Since  Sn{z)  is  continuous  in  z  for  all  finite  values  of  n,  we  have 

I  ASniZo)   1    =    I  Sn{Zo  +  A^)   -  SniZo)   \    <  C  (3) 

*  See  Bromwich,  Theory  of  Infinite  Series,  Art.  81. 


222  INFINITE   SERIES  [Chap.  VI. 

By  combining  (2)  and  (3),  we  obtain 

^  I  ARnizo)  I  +  I  ASnizo)  I 

<  3  e,  I  A2  I  ^  I  A23  I  =  I  Ai2  I  , 

for  all  values  of  Az  equal  to  the  smaller  of  the  increments  Aiz,  A2Z. 
Hence,  since  3  e  is  arbitrarily  small,  |  A/(zo)  1  is  arbitrarily  small  and 
f(z)  is  continuous  at  any  point  Zq  in  the  region  S  of  uniform  con- 
vergence. 

46.  Integration  and  differentiation  of  series.  We  shall  fre- 
quently have  occasion  to  integrate  or  differentiate  a  series  term  by 
term.  The  question  arises  whether  the  resulting  series  represents 
the  integral  or  derivative,  as  the  case  may  be,  of  the  function  defined 
by  the  given  series.  Suppose  a  function  f{z)  is  defined  by  the  rela- 
tion 

f(z)  =  ui{z)  +  Mz)  +  •  •  •  +  Un{z)  +  •  •  •  , 

where  the  u's  are  continuous  functions  in  a  region  S  within  which 
the  given  curve  C  lies.  Denote  by  Sn{z)  the  sum  of  the  first  n  terms 
of  this  series.  The  integral  of  the  function  along  the  path  C,  if  it 
exists,  may  then  be  written 


f  f(z)  dz=    f   L   S„{z)  dz.  (1) 

t/c  <-'C  n=oo 


'c  *Jc 

For  any  definite  value  of  n,  we  may  write 


T  /  M„(0)  dz=    \   ^  Un{z)  dz  =    I  S„{z)  dz. 

I     dc  JC    1  *JC 

Consequently,  the  result  of  term  by  term  integration  of  the  series 
defining  J{z)  may  be  written 

L     fSn{z)dz.  (2) 

n=oo  t/C 

It  can  not  be  assumed  that  the  two  results  (1),  (2)  are  equal.  The 
following  example  furnishes  an  illustration  where  (1)  and  (2)  are  not 
equal. 

Ex.  1.  Given  the  series,  the  sum  of  whose  first  n  terms  is  Sniz)  =  nze~*^. 
Consider  the  term  by  term  integration  of  this  series  where  the  path  C  of  integra- 
tion is  the  X-axis  from  0  to  any  point  /3,  0  <  /3  <  1. 

The  series  converges  for  real  values  of  z.  The  integral  along  the  axis  of  reals 
18  an  ordinary  definite  integral  for  real  values  of  z.    We  have  then 

C^fiz)  dz=   C  L  m^"''dz  =    f^Odz  =  0. 

\  , 


Art.  46.]  INTEGRATION   OF  SERIES  223 

On  the  other  hand,  we  have  J  y^l.** 

L     C^Sn(z)dz  =  L     C^me-^''dz  =  L    1  (i  -  e-"^)  =  1.        ^  1  (e^"^  (^^^ 

n=aa  J  a  n=oo -'o  n  =  oo  ^  *  '*V  J  ^Vv* 

Hence  in  this  case  we  can  not  integrate  the  given  series  term  by  term.  ""   ^  "^ 

We  shall  now  set  up  a  condition  by  means  of  uniform  convergence 
that  will  be  sufficient  for  the  equality  of  (1)  and  (2).  This  condition 
may  be  stated  as  follows: 

Theorem  I.    Let  f{z)  he  defined  by  the  convergent  series 

ui{z)  +  v^(z)  +  •  •  •  +  Un(z)  +  •  •  •  , 

where  Un{z)  is  a  continuoits  function  for  values  of  z  along  an  ordinary 
curve  C.  If  the  series  converges  uniformly  along  C,  we  may  integrate 
the  series  term  by  term,  thus  obtaining 

j  f{z)  dz  =  jui{z)  dz  +  juiiz)  dz -{-•••  -\-  \un{z)  dz  +  •  •  •  .  (3) 


X 


Each  term  of  the  series  is  continuous  and  hence  the  integral 


M„(z)  dz  exists.     Moreover,  since  the  series  converges  uniformly, 

the  function  f{z)  is  a  continuous  function  and  the  integral   i  f{z)  dz 

also  exists.     We  shall  now  show  that  the  relation  given  in  (3)  holds. 
We  may  write 

f{z)  =  Wi(2)  +  u,{z)  +   •  •  •  +  Un{z)  +  Rniz),  (4) 

where  Rn{z)  denotes  the  remainder  of  the  given  series  after  the  first 
n  terms.    By  formula  3,  Art.  17,  we  have 

C  f{z)  dz  =  fluiiz)  +  u,(z)  +  •  •  •  +  Un(z)  +  Rn(z)ldz 
Jc  Jc 

=  fui{z)dz-i-jMz)dz-] -{-Jun{z)dz-\-JRn(z)dz.  (5) 

For  n  sufficiently  large,  say  n  >  m,  the  integral   I  Rn{z)  dz  becomes 
arbitrarily  small.     For,  we  have 

I  fRn(z)dz\^  f\RM)\'\dz\.  (6) 

\Jc  \     Jc 

As  the  series  converges  uniformly,  we  have  for  all  values  of  z  along  C 
\Rn{.z)  I  <  c,        n>  m. 


224  INFINITE  SERIES  [Chap.  VI. 

Hence,  from  (6)  we  obtain 

I  Rn{z)  dz   <  i  I  \  dz  \,        n  >  m 

where  L  denotes  the  length  of  the  path  C  of  integration  and  is  there- 
fore finite.     Since  €  •  L  is  arbitrarily  small,  we  have 


L    \Rn{z)  dz  =  0. 


Consequently,  when  n  is  allowed  to  increase  without  limit,  we 
obtain  from  (5)  the  relation  given  in  the  theorem. 

It  is  necessary  also  to  set  up  some  criterion  for  the  differentiation 
of  a  series  term  by  term;  for,  it  can  not  be  assumed  that  the  series 
formed  by  differentiating  the  various  terms  of  a  given  convergent 
series  is  equal  to  the  derivative  of  the  function  defined  by  that 
series.     The  following  example  furnishes  an  illustration. 


Ex.  2.    Given  the  series 


sin2z      sin  3  z  _  sin  4  2 


The  series  converges  for  real  values  of  the  variable,  and  defines  the  function 
for  values 
term,  we  get 


^  for  values  of  z  lying  between  —v  and  v.     Differentiating  the  series  term  by 


cos  z  —  cos  2  z  +  cos  3  z  —  cos  4  z  +  •  •  •  • 

z  • 

This  series  of  derivatives  does  not  represent  the  derivative  of  ^;  for,  it  does  not 

even  converge  for  values  of  z  other  than  z  =  0.  For,  in  order  that  a  series  shall 
converge,  we  must  have,  as  we  have  seen,  the  Umit  of  the  ?i'*  term  equal  to  zero. 
However,  in  the  case  under  consideration,  the  Umit  L  \  cos  m  \  does  not  even 

n=oo 

exist  for  z  5^  0. 

If  the  terms  of  the  given  series  are  holomorphic  in  a  given  region 
S,  we  have  the  following  theorem,  which  furnishes  a  convenient 
criterion  for  differentiating  or  integrating  term  by  term  such  series  as 
we  shall  have  occasion  to  consider.  It  also  furnishes  a  condition  that 
the  function  defined  by  the  series  shall  be  holomorphic  in  S, 

Theorem  II.     Let  f(z)  be  defined  by  the  convergent  series 
Ui{z)  +  v^{z)  +  •  •  •  +  Un{z)  +  •  •  •  , 


Art.  46.]  DIFFERENTIATION  OF  SERIES  225 

where  Un{z)  is  holomorphic  in  a  region  S.  If  this  series  converges  uni- 
formly in  every  simply  connected  closed  region  lying  wholly  in  S  and 
bounded  by  an  ordinary  curve  C,  then  the  series  may  be  integrated  or 
differentiated  term  by  term  for  values  of  z  in  S.  Moreover,  f{z)  is  holo- 
morphic in  S. 

Since  the  given  series  converges  uniformly  and  each  term  is  con- 
tinuous, it  follows  from  Theorem  I  that  it  may  be  integrated  term 
by  term  along  any  ordinary  curve  C  lying  in  S.  It  is  to  be  noted 
that  as  Un(z)  is  also  holomorphic  in  S,  the  integral  of  each  term  of 
the  series  is  zero,  since  C  is  the  complete  boundary  of  a  simply 
connected  closed  region  lying  wholly  in  S.     We  have  then 


i 


f{z)  dz  =  0. 

Consequently,  by  Theorem  IV,  Art.  20,  /(z)  is  holomorphic  in  the 
given  region  S. 

To  show  that  the  given  series  may  be  differentiated  term  by  term, 
we  proceed  as  follows.     Consider  the  series 

fit)  =  urit)  +  u,(t)  +   •  •  •  +  Unit)  +   •  •  •  ,  (7) 

where  t  takes  values  along  the  closed  curve  C.     This  series  converges 
uniformly,  a  property  that  is  not  destroyed  by  multiplying  the  terms 

of  the  series  by  the  factor  ^ — rr-f — \^>  where  z  is  any  point  within  C. 
We  have  then  >  u.-4rvrS  c^ .  t^  t .  <^  v^ '^  •»-  . 

1        fit)       ^     1        Urjt)  1        U^jt)  1       Unit) 

2Tri  it-zf      27rt  (<  -  z)2  "^  27a  (f  -  z)2  "^  ■"  2 Tri  (<  -  z)^  ■•  * 

Integrating  term  by  term,  we  obtain 

1    r  fit)  dt  ^  1    ruiit)dt      1    r  u^jt)  dt 

2inJc  it  -  zf      2TnJc  it  -  z)^'^  2TnJcit  -  zY 

j^  runiS)dt 

"^  *     '  ^  2TdJcit-zY'^ 

From  Art.  20,  it  will  be  seen  that  the  terms  of  this  series  of  integrals 
are  the  first  derivatives  of  the  terms  of  the  given  series.     We  have 

f'iz)  =  w/(z)  +  u^iz)  +   •  •  •   +  M2'(Z)  +   •  .  .  (8) 

for  any  value  of  z  within  C.     But  C  is  any  closed  curve  in  >S;  hence 
(8)  holds  for  any  values  of  z  in  *S  as  stated  in  the  theorem. 


—    Vx. 


ojL 


226  INFINITE  SERIES  [Chap.  VI. 

Ex.  3.    Given  the  series 

+  fVt^  +    •    •    •   +  ,-S ,^  .o!     ,     ,./o-.     .    ox  +    •   •   •    . 


1.3-5  '  3-5.7   '  5.7.9   '  '   (2n  -  1)  (2n  +  l)(2n  +  3) 

This  series  converges  uniformly  in  any  region  bounded  by  a  circle  about  the 
origin,  which  is  situated  within  the  unit  circle.  It  represents  the  function  *  which 
is  given,  except  for  2  =  0,  by  the  expression 

2V-Z  ^««r=^+3^-i5' 

The  indefinite  integral  of  this  function  is  readily  found  by  integrating  the  given 
series  term  by  term,  thus  obtaining 

'^l-3-5"'"2-3-5.7"''3.5-7-9"'""  "^  n  (2n  -  1)  (2n  +  1)  (2n  +  3)  "^  ' 

Ex.  4.     Given  the  series 

2*  2*  2' 

^  ~  F!  "^  5]  ~  T!  "*■  ■  ■  ■  • 


This  series  converges  uniformly  in  any  finite  region.     The  derived  series  is 

z^       z*       ifi 

i__4-.f _J-  .  .  . 

Consequently,  the  second  series  represents  the  derivative  of  the  function  defined 
by  the  first. 

47.  Power  series.     A  series  of  the  form 

oo  +  ai3  +  0:22^  +    •    •    •    +  OCnZ"  +    •    •    •   , 

where  n  is  a  positive  integer  and 

an  =  fln  +  i&n   =   Pn(C0S  On  +  l  sin  On), 

z  =  X  -\-  iy  =  r(cos  <^  +  i  sin  0), 

is  called  a  power  series  of  complex  terms.  A  more  general  form  of  a 
power  series  may  be  written 

ao  +  ai(z  -  Zo)  +  aiiz  -  2o)^  +   •  •  •   +  «„(«  -  2o)"  +   •  •  •  . 

To  distinguish  the  two  types,  we  may  speak  of  the  second  as  a  power 
series  in  (2  —  20).  For  the  sake  of  simpUcity  we  shall  confine  our 
discussions  for  the  most  part  to  power  series  of  the  first  type.  In 
doing  so  there  is  no  loss  of  generality,  as  power  series  in  0  —  Zo  may 
be  readily  transformed  into  series  of  this  type.  Because  of  their 
importance,  we  shall  consider  some  of  the  special  properties  of  the 
power  series.     Among  these  properties  is  the  following: 

*  Schlomilch,  Ubungsbnch  zum  Studium  der  Hoheren  Analysis,  4"'  Ed.,  Vol.  2, 
p.  239. 


r 


Art.  47.]  POWER  SERIES  .  227 

Theorem  I.     If  for  some  positive  number  G  we  have  far  all  values  of  n 

1  an  I  •  I  Zo"  I  -  G^, 

where  Zo  is  a  constant  value  of  z,  then  San2"  converges  absolutely  for  all 
valves  of  z  for  which  |  2  |  <  |  Zo  |. 

Denoting  the  modulus  of  zq  by  ro,  we  have  by  the  conditions  of 
the  theorem 

P„ro"  =  G. 

The  series  of  absolute  values  may  be  written 

Po  +  Pir  +  •  •  •  +  Pnr"  +  •  •  •  =  po  +  Pi  f  H  ro+  •  '  •  +  pn  f^j  ro"  -I 

^g|i+-+  •  •  •  +(-)"+ ]• 

(         ro  VoJ  ) 

The  series  within  the  braces  converges  to  the  limiting  value  

ro 
if  we  have  r  <  ro.  Consequently,  for  |  0  |  <  |  zo  |  the  series  of 
moduU  converges,  and  hence  the  given  series  SanZ"  converges  abso- 
lutely as  the  theorem  states. 

Ex.  1.     Test  the  convergence  of  the  series 

sinz  +  ;;  sin'^g +Qsin'2  +  •  •  •  +  -  sin^z  +  •  •  •  . 

The  given  series  is  a  power  series  in  sin  z.     If  we  put 

w  =  sin  z, 
we  have 

,   w^  ,   w^   ,  ,   w"   , 

This  series  converges  for  \w\  <  1 ;  for,  we  have  then 

I  ■)/■"  I 
—    <1 

I  ^  I 

for  all  values  of  n.     By  the  foregoing  theorem  the  series  converges  absolutely 
within  the  circle  of  unit  radius  about  the  origin  in  the  PF-plane. 

To  find  the  region  in  the  Z-plane  within  which  the  given  'series  converges,  it 
is  necessary  to  map  upon  the  Z-plane  the  circle  about  the  origin  in  the  TF-plane 
having  the  radius  1  by  means  of  the  relation 

to  =  sin  z. 
We  have  then 

u  +  w  =  sin(x  +  iy)  —  sin  r  cosh  y  -\-i  cos  x  sinh  y, 


228 
whence 


INFINITE  SERIES 


[Chap.  VI. 


«  =  sin  X  cosh  y,    v  =  cos  x  sinh  y. 
The  equation  of  the  circle  in  the  PT-plane  is 

«2  +  t^  =   1. 

Substituting  the  values  of  u,  v  in  terms  of  x,  y,  we  get 

sin*  X  cosh*  y  +  cos*  x  sinh*  y  =  1, 

which  reduces  to  the  form 

cosh  2y  =  cos  2  x  +  2, 
or 

sinh*  y  =  cos*  x. 

The  portion  of  the  fundamental  region  —  -  <  x  =  ^  for  sin  z  bounded  by  the 
curve  given  by  this  equation  is  shown  in  Fig.  83  and  84. 


Fia.  83. 


Fig.  84. 


Theorem  II.  7/  f/ie  'power  series  TianZ""  converges  for  z  =  Zo,  it 
converges  absolutely  for  all  values  of  z  for  which  \  z  \  <  \  Zo  \. 

This  theorem  follows  as  an  immediate  consequence  of  Theorem  I. 
For  if  the  given  series  converges  for  z  =  zo,  then  there  must  exist 
some  positive  number  G  such  that  for  all  values  of  n 

\an\'\zo"\  <G, 

and  consequently  the  series  ^UnZ'*  converges  absolutely  for  values  of 
z  for  which  |  z  |  <  |  Zo  |  as  the  theorem  requires. 
We  have  also  the  following  theorem. 

Theorem  III.  If  the  power  series  San2"  is  divergent  for  z  =  Zi, 
then  it  is  divergent  for  all  values  of  z  for  which  \  z  \  >  \  Zi\. 

By  hypothesis  the  given  series  2Ia„0"  is  divergent  for  z  =  Zi.  It 
must  then  be  divergent  for  all  values  of  z  where  \z  \  >  |  Zi  |;  for,  if 


Art.  47.] 


POWER  SERIES 


229 


it  is  convergent  for  any  such  value  of  z,  say  22,  where  |  22  |  >  1  21  |, 
then  it  must,  by  Theorem  II,  be  convergent  for  all  values  of  z  whose 
modulus  is  less  than  |  22  |  and  therefore  for  z  =  2i,  which  is  a  contra- 
diction of  the  given  hypothesis.  From  the  contradiction  the  theorem 
follows. 

Theorem  II  states  that  if  a  given  series  converges  for  z  =  Zo,  then 
it  converges  within  a  circle  about  the  origin  having  a  radius  equal 
to  I  Zo  I ;  and  Theorem  III  states  that  if  it  is  divergent  for  z  =  Zi,  then 
it  is  divergent  for  all  values  of  z  exterior  to  the  circle  about  the  origin 
whose  radius  is  |  Zi  |.  Nothing  is  said  about  the  convergence  of  the 
series  within  the  region  between  these  two  circles,  or  indeed  upon 
the  circles  themselves,  except  at  the  points  Zo  and  Zi.  The  question 
presents  itseK  as  to  whether  it  is  possible  to  find  a  circle  about  the 
origin  of  radius  R  such  that  the  given  power  series  shall  be  convergent 
for  all  values  of  2  where  j  z  |  <  jB 
and  divergent  for  all  values  of  z 
where  \z  \  >  R. 

It  may  be  shown  as  follows  that 
such  a  circle  of  radius  R  always 
exists,  where  R  may  be  zero,  finite 
and  different  from  zero,  or  infinite. 
Let  as  before  Zo  be  a  point  of  con- 
vergence and  Zi  a  point  of  diver- 
gence of  the  given  series.  Denote 
the  moduli  of  Zo,  Zi  by  po,  pi,  respec- 
tively. Then  we  must  have  po  —  pi. 
If  Po  =  pi,  we  can  take  R  equal  to 
the  conmion  value.    If  po  <  pi,  lay  off  upon  the  X-axis  the  distances  po, 

Pi  +  Po 


Fig.  85. 


pi.     Consider  the  point  ai  = 


For  z  =  ai  the  given  series  is 


either  convergent  or  divergent.  Let  us  suppose  that  it  is  convergent. 
Then  by  Theorem  II  the  series  is  convergent  for  all  values  of  z  within 
the  circle  about  the  origin  whose  radius  is  Oi.  The  region  of  doubt 
lies  now  between  the  two  circles  of  radii  ai,  pi,  respectively.  Consider 
Pi  +  «i 


the  point  02  = 


For  z  =  02  the  series  is  again  either  con- 


vergent or  divergent,  say  divergent.  Then  for  values  of  z  such  that 
I  z  1  >  oa  the  series  is  divergent  by  Theorem  III.  The  region  of 
doubt  now  lies  between  the  circles  of  radii  ai,  02,  respectively. 
Proceeding  in  this  manner  we  shall   obtain  upon  the  X-axis  an 


230  INFINITE   SERIES  [Chap.  VI. 

infinite  sequence  of  intervals  each  lying  in  the  preceding  one. 
Moreover,  the  length  of  the  intervals  has  the  limiting  value  zero. 
These  intervals  therefore  define  a  definite  number  R.  If  we  now 
describe  a  circle  about  the  origin  having  i2  as  a  radius,  we  can  say- 
that  the  given  power  series  converges  for  values  of  z  for  which 
1 2  1  <  72  and  diverges  for  values  of  z  for  which  \z\  >  R.  For 
z  =  R  the  series  may  or  may  not  converge. 

This  circle  whose  radius  is  R  is  called  the  circle  of  convergence 
of  the  power  series,  and  R  is  the  radius  of  convergence.  The  radius 
of  convergence  may  be  equal  to  zero,  in  which  case  the  given  power 
series  converges  for  2  =  0  only,  or  it  may  be  finite  and  different 
from  zero,  or  it  may  be  infinite,  in  which  case  the  given  series  con- 
verges for  all  finite  values  of  z.  Nothing  can  be  said  from  the  dis- 
cussion thus  far  concerning  the  convergence  of  the  series  for  points 
on  the  circle  of  convergence  itself.  As  a  matter  of  fact  a  power 
series  may  converge  absolutely  at  every  point  on  the  circle  of  conver- 
gence, or  it  may  converge  conditionally  at  every  such  point,  or  it  may 
converge  conditionally  at  certain  points  upon  this  circle  and  diverge 
at  other  points,  or  finally  it  may  diverge  at  all  points  upon  this  circle.* 

We  shall  need  methods  by  which  we  may  determine  the  radius 
of  convergence  of  a  given  power  series.  It  evidently  depends  upon 
the  coefficients  of  the  given  series.  A  relation  between  the  radius  of 
convergence  and  the  coefficients  of  the  given  series  is  given  by  the 
following  theorem. 

Theorem  IV.     If  the  coefficients  of  the  given  power  series  Sa„z"  are 


such  thai  the  limit    L 


OCn+l 


exists,  then  the  value  of  this  limit  is  equal 

to  the  reciprocal  of  R;  that  is,  it  is  the  reciprocal  of  the  radius  of  con- 
vergence of  the  given  series. 

Put 

«n+l 


L 

n=oo 


«n 


^A. 


To  prove  that  -j  is  equal  to  the  radius  of  convergence,  it  is  neces- 
sary to  show  that  the  given  power  series  converges  for  all  values  of  z 
where  |  2  |  <  -j  and  diverges  for  all  values  of  z  where  |  2  |  >  -j  • 

*  See  EncydopMie  des  Sci.  Math.,  II7,  p.  15. 


Art.  47.]  POWER. SERIES  231 

We  may  readily  show  that  the  power  series  converges  for  values 
of  z  where  |  2  |  <  -j .    As  in  the  demonstration  of  Theorem  I,  let 

OLn  =  Pn (cos  dn  +  l  SIR  dn),       Z  =  r(cOS  (f)  -{-  l  SID.  (f>) . 

By  Theorem  III,  Art.  42,  the  given  power  series  converges  if  the 
series  of  moduli 

Po  +  Pir  +  pzr^  +  •  •  •  +  pnT""  +  •  •  •  (1) 

converges.     This  series  converges  if  we  have 

n=oo       PtJ"  n=oo     Pn 

By  the  condition  of  the  theorem  we  have 


L  ^^=  L 

n=a>     Pn  n=oo 


Hence  the  condition  that  (1)  converges  is  that 

rA  <  1; 

that  is,  \z  \  ^  r  <  -J- 

Consequently,  the  given  series  converges  for  all  values  of  z  within 

the  circle  of  radius  -;  • 
A 

The  given  series  likewise  diverges  for  all  values  of  z  without  the 

circle  of  radius  -j-     To  show  this,  suppose  it  to  converge  for  some 

value  Zo  without  this  circle.  Let  Zi  be  any  point  outside  of  the 
circle  such  that  \  Zi\  <  \zo\.  Then  by  Theorem  II  the  given  series 
converges  absolutely  for  zi.    We  have  then  the  convergent  series 

PO  +  Pin  +  •  •  •  +  Pnri"  +  •  •  •  .  (2) 


However,  we  have 


n  =  QO         Pnfl 


since  ri  >  -j  •  This  result  contradicts  the  conclusion  that  series  (2) 
is  convergent.  From  this  contradiction  it  follow^  that  San^"  cap 
not  converge  for  any  value  of  z  exterior  to  the  circle  of  radius  -j* 


232  INFINITE  SERIES  [Chap.  VI. 

Since  the  given  series  converges  for  all  values  of  z  within  the 
circle  about  the  origin  having  the  radius  -7  and  is  divergent  for  all 

values  of  2  without  this  circle,  it  follows  that  -j  must  be  equal  to  R, 

the  radius  of  convergence,  as  the  theorem  states. 

The  application  of  Theorem  IV  to  the  problem  of  determining  the 

radius  of  convergence  depends  upon  the  existence  of  the  limit  L  -^^  \  • 

The  theorem  gives  a  sufficient  but  not  a  necessary  condition  for 
convergence.  There  are  convergent  series  for  which  this  limit  does 
not  exist.     The  following  series  furnishes  an  illustration. 

Ex.  2.     Given  the  series 

1  -L  i  2  -I L.  z2  J I »3  J_ z4  4.  .  .  .     I t ~i  n— 1  _| t z2n  I     .   .   . 

^22-3  22.3      "22-32  ~2"'3'^i  '2" -3" 

This  series  is  convergent  f or  |  2  |  <  1;  for  putting  2  =  1,  we  get  a  series  whose 
terms  are  less  than  the  corresponding  terms  of  the  convergent  series  ^(  ^ )  • 
an+i 


does  not  exist  since 


oscillates  between  -    and  - , 
2  3 


The  limit  L 

n=oo 

depending  upon  whether  an  even  or  odd  term  is  taken  as  the  n'*  term. 

The  following  theorem  *  gives  us  a  means  of  determining  a  radius 

of  convergence  that  is  applicable  to  any  power  series. 

00 

Theorem  V.     Given  the  series  ^anZ";  and  let  Pn=  \ocn\.    If  A 

n=0 

is  the  maximum  limit  of  the  sequence 

Pi,  v^,  v^,  .  .  .  ,  \/p„,  .  .  .  ,  (3) 

then  J  is  equal  to  the  radius  of  convergence  of  the  given  series. 

By  the  maximum  limit  of  a  sequence  is  understood  the  largest 
number  that  can  be  obtained  as  the  limit  of  a  subsequence  of  the 
given  sequence.  In  the  particular  case  under  discussion  we  are  to 
consider  the  various  subsequences  that  may  be  selected  from  (3)  and 
denote  by  A  the  largest  number  that  can  be  obtained  as  the  limiting 
value  of  any  of  these  subsequences. 

*  This  theorem  was  first  demonstrated  by  Cauchy.  See  his  Analyse  Alg., 
p.  286,  also  Enq^clopMie  des  Sci.  Math.,  II7,  p.  6. 


Akt.  47.] 


POWER  SERIES 


233 


We  wish  to  show  that  the  given  series  converges  for 

1 


z\  < 


A' 


that  is,  within  the  circle  C  (Fig.  86),  having  the  origin  as  a  center 
and  i2  =  J  as  a  radius.     Let  z'  be  any  point 
within  the  circle  C.    We  have  then 


2'       = 


A  + 


,  where  0  <  e. 


There  are  at  most  a  finite  number  of  ele- 
ments of  the  sequence  (3)  greater  than  or 
equal  to  A  -\- 1.  Suppose  m  is  the  largest  of 
the  subscripts  of  these  elements.  We  have 
then 

1  n/— 

■j— 7-7  =A+e>  Vpn,         n>  m, 


Fig.  86. 


or  \z'  \  Vpn  <  1,        n>  m. 

We  therefore  have 

Pn  I  2'"  I  =  I  an2'"  1  <  1,       n>m. 

00 
It  follows  from  Theorem  I  that  the  series     ^    UnZ"",  and  hence  the 

given  series,  converges  absolutely  for  all  values  of  z  within  the  circle 

whose  radius  is  .        •     As  e  is  arbitrarily  small  it  follows  that  the 

series  2Iq:„2"  converges  absolutely  within  the  circle  C. 
We  wish  now  to  show  that  the  given  series  diverges  for 

Let  z"  be  any  point  exterior  to  the  circle  C.    We  have  then 

1 


z"     = 


A- 


>0. 


There  are  now  an  infinite  number  of  elements  of  the  sequence  (3) 
greater  than  A  —  e;  that  is,  for  an  infinite  number  of  values  of  n 
we  have 


=  A 


<  \^, 


Pn, 


234  INFINITE  SERIES  [Chap.  VI. 

or 

\zf'\^n>l. 

Then  for  |  2  |  =  \z"  \  we  have,  for  an  infinite  number  of  values  pf 
n, 

Pn  I  2"  I    =    I  «n2"  1    >   1. 

Consequently,  the  given  series  SanZ"  can  not  converge  for  |  z  |  >  -j  • 

Since  San^"  is  convergent  for  all  values  of  z  lying  within  the 

circle  of  radius  -j  and  divergent  for  all  values  of  z  lying  without  this 

A 

circle,  it  follows  that  -^  is  equal  to  R,  the  radius  of  convergence, 
A. 

which  estabhshes  the  theorem. 

Whenever  the  sequence  (3)  has  a  definite  limit  as  n  becomes  in- 
finite, the  various  subsequences  have  the  same  limit  and  hence  the 
maximum  limit  is  the  limit  of  the  sequence.     It  will  be  observed 

""■^^    have 


also  that  whenever  both  the  sequence  (3)  and  the  ratio 

a  limit,  the  two  limits  are  the  same,  since  both  are  the  reciprocal  of 
the  radius  of  convergence.  Theorem  V  often  enables  us  to  de- 
termine the  radius  of  convergence  even  if  the  sequence  (3)  has  no 
definite  limiting  value. 

Ex.  3.     Determine  the  radius  of  convergence  of  the  p>ower  series  given  in  Ex.  2. 
We  have  the  sequence  of  positive  values 

i^\ lWW^''sJWW 

The  limit  of  the  subsequence  in  which  the  odd  roots  alone  are  taken  is 

1 

The  limit  of  the  subsequence  in  which  the  even  roots  alone  are  taken  is 


i.NWir=i.(r 


1 


Art.  47.]  POWER   SERIES  235 

No  other  subsequence  has  a  different  limit  and  hence  the  sequence 

Pi,     Vp2,    Vp3,     .     .     .    ,     Vp„,     .     .     . 

has  the  limit  — = ,  and  the  radius  of  convergence  of  the  given  power  series  is  v^. 
V6 

The  following  theorem  is  of  importance  in  the  discussion  of  ana- 
lytic functions. 

Theorem  VI.  The  power  series  IIa:„2"  converges  uniformly  in  the 
closed  region  bounded  by  any  circle  about  the  origin  whose  radius  is 
R'  <  R,  where  R  is  the  radius  of  convergence  of  the  given  series.  In 
the  open  region  bounded  by  the  circle  of  convergence  the  power  series 
represents  a  function  which  is  holomorphic. 

Let  R"  be  any  number  such  that 

R'  <  R"  <  R. 

Then  the  series  of  positive  terms 

\a,\  +  \a,\R;'+\a,\R^'^   •  •  •  (4) 

converges.  For  yalues  of  z  such  that  \z\  =  R'  the  terms  of  the 
given  series  are  less  iii'absolute  value  than  the  corresponding  terms 
of  (4).  Hence,  by  Theorem  I,  Art.  45,  the  given  series  converges 
absolutely  and  uniformly  in  the  closed  region  bounded  by  the  circle 
of  radius  R'. 

Since  any  closed  region  bounded  by  an  ordinary  curve  C  and 
lying  wholly  in  the  open  region  bounded  by  the  circle  of  convergence 
can  be  included  within  a  circle  of  radius  R'  <  R,  it  follows  that  the 
given  power  series  is  absolutely  and  uniformly  convergent  in  every 
such  closed  region.  Therefore,  by  Theorem  II  of  the  last  article 
the  given  series  represents  an  analytic  function  in  the  open  region 
bounded  by  the  circle  of  convergence,  as  stated  in  the  theorem. 

The  foregoing  theorem  states  nothing,  however,  as  to  the  uniform 
convergence  of  the  given  power  series  in  the  open  region  bounded  by 
the  circle  of  convergence. 

Ex.  4.     Consider  the  convergence  of  the  series 


The  circle  of  convergence  has  the  radius 


R  =  \rT=  =  2. 


236  INFINITE  SERIES  [Chap.  VI. 

The  remainder  Rn{z)  after  the  first  n  terms  is 

2" 


Rniz)    = 


2»-i(2  -  z) 


As  \  Rn(z)  I  for  any  value  of  n  may  be  made  as  large  as  we  choose  by  taking  |  z  \ 
suflBciently  near  2,  it  follows  that  there  is  no  number  m  independent  of  z,  such  that, 
for  all  values  of  z  within  the  circle  of  convergence, 

I  Rniz)  I  <  «,        n>  m. 

Hence  the  series  does  not  converge  xmiformly  in  the  open  region  within  the  circle 
of  convergence. 

However,  for  all  values  of  z  within  a  circle  about  the  origin  having  a  radius 
R'  <  2,  there  exists  a  number  m  such  that  for  n  >  rw,  we  have 


Rniz) 


2n-l(2  -  z) 


<e, 


Consequently  in  the  closed  region  bounded  by  a  circle  of  radius  R'  <  R  the 
given  series  converges  uniformly. 

The  following  theorem  gives  a  condition  under  which  a  power 
series  is  uniformly  convergent  in  the  closed  region  bounded  by  the 
circle  of  convergence. 

Theorem  VII.  If  IIa„3"  is  absolutely  convergent  at  one  point  on 
the  circle  of  convergence,  then  it  converges  absolutely  and  uniformly  in 
the  closed  region  bounded  by  the  circle  of  convergence. 

If  the  given  series  is  absolutely  convergent  at  one  point  on  the 
circle  of  convergence,  say  at  z  =  Zq,  we  know  from  the  definition  of 
absolute  convergence  that  the  series  2  |  a„  1  i2"  converges,  where  R 
is  the  radius  of  convergence.  However,  any  point  on  the  circle 
whose  radius  is  R  gives  the  same  series  of  moduli.  Hence,  for 
values  of  z  such  that  \z\  =  R  the  terms  of  the  given  series  are  not 
greater  in  absolute  value  than  the  corresponding  terms  of  S  |  a„  |  i2". 
Hence  by  Theorem  I,  Art.  45,  the  given  series  converges  absolutely 
,  and  uniformly  in  the  closed  region  bounded  by  the  circle  of  radius  R. 

Ex.  6.     Given  the  series  ^-k^  ,  the  character  of  whose  convergence  is  to  be 

examined. 

This  series  is  absolutely  convergent  for  z  =  1,  since  the  series  of  moduli  2)k^  is 

convergent.  Hence,  by  Theorem  VII  the  series  converges  absolutely  and  uni- 
formly in  the  closed  region  bounded  by  the  circle  about  the  origin  whose  radius 
is  1. 


Akt.  47.]  POWER  SERIES  237 

The  function  represented  by  this  series  (Theorem  II,  Art.  45)  is 
continuous  in  the  closed  region  bounded  by  the  unit  circle,  and  by 
Theorem  VI  is  holomorphic  within  this  circle.  This  particular  func- 
tion,* however,  is  not  holomorphic  upon  the  unit  circle  itself. 

Corollary.  If  T,otnZ'^  is  divergent  or  conditionally  convergent  at 
any  point  on  the  circle  oj  convergence,  then  it  can  he  at  best  only  condi- 
tionally convergent  at  any  other  point  on  this  circle. 

This  proposition  follows  as  a  consequence  of  Theorem  VII;  for, 
if  the  given  series  is  absolutely  convergent  at  any  other  point  on  the 
circle  of  convergence,  then  by  Theorem  VII,  it  must  converge  abso- 
lutely for  all  values  of  z  for  which  \z\  —  R,  the  radius  of  convergence, 
and  this  is  a  contradiction  of  the  hypothesis.  Hence,  if  the  given 
series  converges  at  any  other  points  on  the  circle  of  convergence,  it 
must  converge  conditionally. 


Ex.  6.     Consider  the  character  of  the  series 


Xf' 


This  series  is  conditionally  convergent  at  z  =  —  1.  It  is  divergent  at  z  =  1. 
Hence,  in  this  case,  the  series  can  not  be  absolutely  convergent  at  any  point  on 
the  unit  circle.  * 

For  the  differentiation  and  integration  of  a  power  series  we  have 
the  following  theorem. 

Theorem  VIII.  The  power  series  HanZ'^  may  he  differentiated  or 
integrated  term  hy  term  in  the  open  region  hounded  by  the  circle  of  con- 
vergence. The  circle  of  convergence  of  the  resulting  series  is  the  same 
as  that  of  the  given  series. 

That  the  given  power  series  may  be  integrated  or  differentiated 
term  by  term  in  the  open  region  bounded  by  the  circle  of  conver- 
gence follows  from  Theorem  II  of  the  last  article  by  the  same  reason- 
ing as  was  employed  in  the  demonstration  of  Theorem  VI. 

'The  resulting  series  in  either  case  has  the  same  circle  of  conver- 
gence as  the  original  series.  We  shall  show  this  to  be  true  for  term 
by  term  differentiation.  A  similar  argument  will  establish  the  truth 
of  the  statement  for  term  by  term  integration.     The  series  of  deriva- 

•tives 

f'{z)  =  ui'{z)  +  u,'iz)  +  •  •  •  +  uJiz)  +  .  .  •  (5) 

is  a  power  series.     By  the  first  part  of  the  theorem  under  considera- 
tion the  series  (5)  converges  for  all  values  of  z  within  the  circle  of 
radius  R.    We  must  show  that  it  is  divergent  for  values  of  z  exterior 
♦  See  Picard,  Traile  d'arudyse,  2^  Ed.,  Vol.  2,  p.  74. 


238 


INFINITE  SERIES 


[Chap.  VI. 


to  the  circle  of  radius  R.  Suppose  it  should  converge  for  some  value 
Zi  exterior  to  this  circle.  Then  for  values  of  z  such  that  |  2  1  <  R", 
where  R  <  R"  <  \zi\,  the  series  (5)  converges  uniformly  and  can 
be  integrated  term  by  term.  As  a  result  of  integration  we  should 
have  the  original  power  series,  except  as  to  an  additive  constant, 
and  this  power  series  must  then  converge  for  all  values  of  z  such 
that  1  z  I  <  R"-  This,  however,  is  impossible  since  values  of  z  ex- 
terior to  the  circle  of  convergence  of  the  given  series  are  thus  included. 
From  this  contradiction  it  follows  that  the  series  (5)  can  not  con- 
verge for  values  of  z  exterior  to  the  circle  of  radius  R.  Since  the 
series  (5)  converges  for  all  values  of  z  within  the  circle  of  radius  R 
and  diverges  for  all  values  of  z  exterior  to  it,  it  follows  that  R  is  the 
radius  of  convergence  of  (5)  as  stated. 

48.  Expansion  of  a  fimction  in  a  power  series.  We  have  seen 
that  in  the  open  region  bounded  by  the  circle  of  convergence,  a 
power  series  represents  a  function  which  is  holomorphic.  We  shall 
now  show  that  a  function  may  be  uniquely  represented  by  a  power 
series  in  the  neighborhood  of  any  point  of  a  region  in  which  it  is 
holomorphic.  Moreover,  we  shall  develop  a  method  for  obtaining 
the  required  power  series.  The  results  may  be  stated  in  the  follow- 
ing theorem. 

Theorem  I.  If  fiz)  is  holomorphic  in  a  given  region  S,  then  in  the 
neighborhood  of  any  point  Zq  in  S,  f{z)  can  he  represented  by  a  power 
series  in  (z  —  zq),  and  thai  in  one  and  only  one  way,  namely: 


f{z)=f{zo)-\-f'{^o)(z- 


■Zo)  +  ^^iz-Zoy-{- 


+  ■ 


(z-Zo)"+ 


Let  zo  be  any  point  in  the  given 
region  S.  About  the  point  Zo  as  a 
center  draw  the  circle  C,  having  the 
radius  r  and  lying  within  S.  Let  the 
complex  variable  t  take  values  corre- 
sponding to  the  points  of  C.  For  any 
point  z  within  the  circle,  we  have  then 


Fig.  87. 


\z  —  Zo\<\t  —  Zq\,    or 
Consider  now  the  series 


Z—  Zo 


t  —  Zq 


z-  Zo    _^  (z  -  ZoY 


t-Zo  '   (t-ZoY  '   {t-  zoy 


I       (g  -  2o)"       , 


<  L 


(1) 


Art.  48.]  EXPANSION   OF   FUNCTIONS  239 

This  series  converges  for  values  of  z  within  C  and  represents  the 
function ;  for,  it  is  a  geometric  series  having  the  ratio  ;; ^  • 

t  —  Z  t  —  Zq 

Considered  as  a  series  in  the  complex  variable  t,  it  converges  uni- 
formly upon  the  circle  C;  for,  |  z  —  2:0  |  and  |  f  —  20  1  =  r  are  then 
both  constant  and  the  series  of  maximum  numerical  values 

1        \z-  Zo\    ,    \z  -  Zo\^  I  z  -  2o  I" 


r  r" 


rn+l 


converges.  The  uniformity  of  the  convergence  is  not  disturbed  if  we 
multiply  each  term  hy  f{t).    We  thus  obtain 

m  ^  m    {z-z,)m    {z-z,ym 

t-Z        <-2o  it-ZoY      "^       {t-Z^f 

-^  '"  ^   {t-  z,Y^^  +  '"  '  (2). 

Since  this  series  converges  uniformly,  we  may  integrate  it  term  by 
term  with  respect  to  t,  the  integral  being  taken  around  the  circle  C. 
We  thus  obtain 

j_  n{t)dt _  .1    rmdt  1    r  mat 

^^^       2xiJ    t-Z      27rtJc<-0o       ^         '''2TriJc{t-ZoY 
Each  of  these  integrals  is  a  constant,  and  we  have  by  Art.  20, 

/(»)(2o)_  1   r  mdt        ^  =  012 

Replacing  the  integrals  in  (3)  by  their  equivalent  values  in  terms  of 
the  successive  derivatives  of  f{z),  we  have  for  z  =  Zo  the  required 
expansion  known  as  Taylor's  series,  namely: 

fiz)  =f(zo)  -\-fizo){z  -  Zo)  -\-^^(z-Zo)^ 

+  ••  •  +^-^(«-^o)"+   •  ••.  (4) 

For  2o  =  0  we  have  for  the  expansion  of  the  given  function  in  the 
neighborhood  of  the  origin  a  special  form  of  Taylor's  series  known  as 
Maclaurin's  series,  namely: 

f{z)  =/(0)  -\-f\0)z+Qylz''  +  •  •  •  -{-CMz--\-  ....      (4')  \ 


240  INFINITE  SERIES  [Chap.  VI. 

Within  the  circle  C,  that  is  in  a  neighborhood  of  zo,  the  given 
function  f{z)  can  therefore  be  represented  by  a  power  series.  More- 
over, within  C  the  given  function  can  be  represented  by  a  power 
series  in  {z  —  Zo)  in  only  one  way.     For  convenience  put 

n!     """• 

The  series  (4)  can  then  be  written 

oo  +  ai  (z  -  2o)  +  aziz  -  Zo)^  +   •  •  •   +  an(z  -  Zo)"  +   •  •  •  .     (5) 

Suppose  it  is  possible  that  within  a  circle  Ci,  whose  radius  is  equal  to 
or  less  than  that  of  C,  f{z)  can  be  represented  by  a  second  power 
series,  say 

/(2)  =  i8o  +  /3i(z-Zo)+/32(z-Zo)2+  •  •  •  +/3„(z-Zo)"+  •  •  •  .  (6) 
Subtracting  (6)  from  (5)  we  have 

0  =  (oo  -  /3o)  +  (ai  -  /3i)  (z  -  Zo)  +  ("2  -  P2)  (z  -  Zo)' 

+  •  •  .  +  (a„  -  ^„)  (z  -  Zo)»  +  •  •  •  .  (7) 

This  relation  holds  for  all  values  of  z  within  Cf,  hence  for  z  =  zo. 
Putting  z  =  Zo,  we  get 

0  =  Oo  —  /3o,     or     Oo  =  i3o. 

For  z  5^  Zo,  however,  we  have 

0=(ai-^i)  (z-Zo)  +  (a2-/32)(z-Zo)2H +(an-^n)  (z-Zo)"+  •  •  •  . 

Dividing  by  (z  —  Zo),  we  obtain 

0  =  (ai-/3i)  +  (a2-/32)(z-Zo)+  •  •  •  +  (a„  -  ^„)  (z- Zo)"-^+  •  •  •  . 

This  series  converges  uniformly  within  or  upon  any  circle  about 
Zo  as  a  center  and  lying  within  Ci.  Hence  it  defines  a  continuous 
function,  say  (f>(z).  Since  </>(z)  is  continuous  and  equal  to  zero  for 
z  7»^  Zo,  we  have  for  z  =  Zo, 

<f>(zo)  =  L  (t>{z)  =  L  0  =  0. 

Consequently,  we  have 

Q  =  <^(zo)  =  ai  —  /3i  =  0,     or     ai  =  /3i. 

Continuing  in  this  manner  we  may  show  that 

an  =  j8n,         n  =  0,  1,  2,  .  .  .  ; 


/'(O)  = 

1, 

/"(O) 
2!    ~ 

1, 

3!     ~ 

1, 

f^"\0) 

1 

Art.  48.]  EXPANSION  OF  FUNCTIONS  241 

hence,  in  the  neighborhood  of  Zo  the  given  function  can  be  represented 
by  a  power  series  in  {z  —  Zo)  in  only  one  way.  Since  Zo  is  any  point 
in  the  given  region  S,  the  theorem  is  established. 

Ex.   Expand  f{z)  =  ^  _    in  a  Maclaurin's  series. 
We  have 

m  =  0, 

/'(z)  =  (1  -  ^)-^ 
f'iz)  =  2  (1  -  z)-3, 

r"iz)=2'S{l-z)-\ 

n  I 
Hence,  in  the  neighborhood  of  the  origin  the  series 

Z  +  Z2  +  Z»  +    •    •    •    +  2"  +    •    •    •  (8) 

represents  the  given  function. 

It  is  to  be  noted  that  the  power  series  in  (z  —  zo)  arising  by  Tay- 
lor's expansion  of  a  given  function  which  is  holomorphic  in  a  region  S 
represents  that  function  for  all  values  of  z  within  any  circle  that 
can  be  drawn  about  the  given  point  Zq,  so  long  as  it  hes  within  the 
given  region  S  and  incloses  only  points  of  S;  for,  it  is  clear  that  any 
such  circle  can  be  selected  as  the  curve  C  along  which  the  inte- 
grals are  taken  that  determine  the  coefficients  in  the  expansion. 
[Moreover,  if  as  in  the  illustration  given  above,  the  function  is  analytic 
within  the  entire  circle  of  convergence,  the  series  represents  the 
function  giving  rise  to  it  within  the  whole  of  that  circle. 

We  now  have  shown  that  every  power  series  defines  a  function 
which  is  holomorphic  in  the  open  region  bounded  by  the  circle  of  con- 
vergence, and  conversely  that  a  function  can  be  expanded  in  a  power 
series  in  the  neighborhood  of  any  point  in  the  region  S  in  which  it 
is  holomorphic.  It  will  be  seen,  therefore,  that  power  series  play  an 
important  role  in  the  discussion  of  analytic  functions.  Indeed, 
Weierstrass  based  his  entire  development  of  the  theory  of  analytic 
functions  upon  power  series. 


242  INFINITE  SERIES  [Chap.  VI. 

EXERCISES 

1.  Determine  the  circle  of  convergence  of  the  series 
(a)  l  +  4z  +  92»  +  162'+  •  •  •  , 

(c)   l+z  +  |^  +  f,+  ..., 

(^)i+x  +  x^  +  x-3+---- 

2.  Discuss  the  uniform  convergence  of  the  series 

^«>i+x  +  xi  +  x3+---+r»+---' 

(^>i+-x  +  x^+x-3+---+r»+---' 

where  |  X  |  >  1.     What  can  be  said  of  the  function  represented  by  the  second 
series  that  can  not  be  said  of  the  function  represented  by  the  first? 
^    -J   3.   Discuss  the  behavior  of  the  series 

for  values  of  2  upon  the  circle  of  convergence,* 

(a)  for  m>  0, 
(6)  for  m  ^  -  1; 
(c)  for  -1  <OT  =  0. 

4.  Show  that  the  two  series  ■* 

1  1-j      1_^  (1  -  z)'  ,  l'3-5  (1  -  zy 
^^2      1      "^2.4        2       "^2.4.6        3        •^•••' 
„     1  g-  1       Ij^  (z  -  1)»      1'3'5  (z  -  1)» 
^2      1      "^2.4        2       "'"2.4.6        3       +••• 

have  the  same  region  of  convergence. 

6.   Determine  the  region  of  convergence  of  the  series 
^ 

z 


'^kM'^liri'h 


Find  the  derivative  of  the  function  represented  by  the  given  series. 
6.   Given  the  series 

1  z  z- 


1.2.3  '  2.3.4  '  3.4.5 

Verify,  by  testing  the  first  and  second  derived  series,  that  they  have  the  same 
circle  of  convergence  as  the  original  series. 

•  See  Goursat,  Coura  d^ analyse  malhematique,  2d  Ed.  (1911),  Vol.  2,  p.  43. 


Akt.  48.]  EXERCISES  243 

7.  Given  the  series 

cos' 2  ,   cos'z  ,   (—  l)"-^cos'*-i«    , 

COS  z 5 1 z •  •  •   H -^ -. h  •  •  •  . 

3  o  2n  —  1 

For  what  valuer  of  cos  z  is  the  series  convergent?  Determine  the  corresponding 
region  in  the  Z-plane.  Does  the  series  represent  a  continuous  function  of  z  in 
this  region? 

8.  Given  the  series 

z^  2^ 

^  1  +  2*  ^  (1  +  2^)2  ^ 

Show  that  this  series  converges  for  all  finite  values  of  2  outside  the  lemniscate 

p2  =  -2  cos  2  0. 

Show  that  this  series  diverges  at  all  points  different  from  zero  within  and  upon 
this  lemniscate. 

9.  Derive  the  following  expansions  and  determine  in  each  case  the  region  of 
convergence : 

(a)  e^  =  i+,  +  ^  +  |i+..., 

(6)  sin 2  =  2  -  g-j  +  g-j  -  ^j  +  •  •  •  , 

(c)  «°«^  =  l-f!  +  4l~6T^  ■  *  •  ' 

(rf)  log(l+2)=2-|  +  |-|+  •  •  .  , 

(e)        (l  +  2)'"  =  l+m2  +  "'^'"^7^^g^+  •  •  •  . 
10.     Derive  the  expansion 

—   =  1  —  2  +  2"  -    •    •    • 

1+2  ^ 

and  verify  the  expansion  in  Ex.  9  (rf)  by  integration  of  this  series.  Derive  in 
a  similar  manner  the  expansion 

iz     _     _  2*  ,  2^ 
+  2^  "  ^       3  "^  5 


arc  tan  z  =    j        "*^_,  =  «  —  o  +  I — 


11.   Making  use  of  the  expansions  in  Ex.  9,  derive  the  following  relations: 
(o)  cos  2  2  =  cos*  z  —  sin^  z, 

(6)  log(^)  =  log  1  -  log  (1  -  2), 

1  2 

(c)  tan  2  ==  2  +  2  23  +  —  2^  +  •  •  .  , 

(d)  sinh2=  — 2 — '  "=  ^  "^  3I  "^  Sl  "^  '  '  '  ' 

(e)  cosh  z  =        ^     '"^'^r!'^4l+"'**  % 
Verify  these  results  by  Taylor's  theorem. 


244  INFINITE  SERIES  [Chap.  VI. 

12.   From  the  expansion 


derive  the  expansion 


tan«  =zH-g2»  +  Y^2*  + 


11  1  ^. 


IS.  Making  tise  of  the  expansions 

z'    ■    z"        Z' 
3~!  "^  5l  ~  f ! 


8inz=z-3-,  +  5^-7^  + 
^       z^       ^ 


derive  the  expansions 

^  (a)  C8CZ  =  i  +  ^j  +  3^  +  •  •  •  ,        yyot  <y   r>^>-^ 

(6)  sec«  =  1+1^  +  1^+  ..•  . 

14.   Derive  the  expansions 

(  \  ■  r'       dz  ,   1  z3       1 . 3  gs 

fh\        C  ^dz        Z^   ,        Z*       ^      52*      . 

^^^      J   ^^=  2  +  4721  +  674!+  •••• 

16.  Verify  the  formulae 

-J  sin  (a  +  /3)  =  sin  a  cos  iS  4-  cos  a  sin/3, 
/  cos  (a  +  /3)  =  cos  a  cos  /3  —  sin  a  sin  /8 

by  means  of  the  power  series  expansion  of  the  sine  and  cosine. 
16.   Given  the  expansion 

^'=l+^  +  2!  +  3!+---- 
By  aid  of  this  series  prove  that 

and  give  the  reason  for  each  step. 


CHAPTER  VII 
GENERAL  PROPERTIES  OF  SINGLE-VALUED  FUNCTIONS 

49.  Analytic  continuation.  In  the  present  chapter  we  shall  dis- 
cuss some  of  the  general  properties  of  single-valued  functions.  Let 
us  first  consider  how  a  function  which  is  holomorphic  in  a  certain 
region  may  be  completely  represented  in  that  region  by  means  of 
power  series. 

Consider,  for  example,  the  function 

Expanding  this  function  in  a  Maclaurin  series,  we  have 

1  +  0  -I-  22  +  03  +    •    •    •    +  0»  +     •    •    •    .  (1) 

This  series  converges  within  the  circle  C  of  unit  radius  about  the 
origin.  Since  the  given  function  f{z)  is  holomorphic  for  all  finite 
values  of  z,  except  z  =  1,  it  follows  from  Art.  48  that  the  series  (1) 
represents  that  function  for  all  values  of  z  within  C.  However,  for 
values  of  z  exterior  to  C  the  series  (1)  does  not  converge  and  hence 
can  not  be  said  to  represent  the  given  function.  We  may  for  con- 
venience denote  the  aggregate  of  functional  values  corresponding  to 
values  of  z  within  C,  as  given  by  the  series  (1),  by  the  symbol  ^(z). 

We  shall  speak  of  <^(z)  as  an  element  of  the  function  f{z)  =  :; 

1  —  z 

A  general  definition  of  an  element  of  an  analytic  function  wUl  be 
given  later  in  this  article.  Since /(z)  can  be  represented  by  a  Taylor's 
expansion  in  the  neighborhood  of  any  point  of  a  region  in  which  the 
function  is  holomorphic,  there  is  similarly  an  element  0o(z)  correspond- 
ing to  any  finite  point  Zo  of  the  complex  plane,  except  the  point  z  =  1. 
The  power  series  defining  these  respective  elements  of  /(z)  converge 
within  circles  which  may  overlap. 

For  example  let  zo  be  a  point  within  C,  so  selected  that  for  some 

values  of  z  upon  C  we  have  \z  —  zq\  <  ' — tt-^^'    Expanding  the  given 

function     _     in  powers  of  (z  —  zo),  that  is  in  a  Taylor's  series,  we 

245 


246  SINGLE-VALUED   FUNCTIONS  [Chap.  VII. 

get 


1  -  20    '     (1  -  20)^    '     (1  -  20)'    '  '        (1  -  20)" 

This  series  is  a  geometric  series  having  the  ratio  z ,  and  hence  it 

1  —  2o 

converges  for  values  of  z  such  that 

l2-2o|<ll-2o|; 

that  is,  it  converges  for  values  of  2  within  a  circle  of  Co  of  radius 
1 1  —  2o  1  about  2o  as  a  center.  Since  the  point  20  was  so  selected  that 
at  least  one  point  on  C  is  closer  to  20  than  one-half  of  the  distance 
1 1  —  2o  I,  it  follows  that  Co  must  intersect  C. 

In  that  portion  of  the  plane  included  within  these  two  circles  of 

convergence,  the  given  function     _     is  represented  by  either  of  the 
•  1  —  2 

two  series,  each  giving  the  same  numerical  value  for  any  particular 

value  of  2  within  the  common  region.     Consider  now  an  assemblage 

of  power  series  obtained  from :; ,  such  that  the  corresponding  circles 

1  —  2 

of  convergence  cover  the  entire  finite  portion  of  the  plane  except  the 
one  point  2  =  1,  which  is  not  a  regular  point  of  the  given  function. 
This  assemblage  of  power  series  may  be  said  to  completely  represent 
the  given  function. 

In  the  foregoing  illustration  the  function  f(z)  is  given  by  means 
of  an  algebraic  expression  in  2,  from  which  the  value  of  the  function 
can  be  computed  for  any  value  of  2  except  2=1.  From  this  ex- 
pression we  are  able  to  obtain  an  expansion  of  the  function  in  a 
power  series  in  the  neighborhood  of  any  point  of  the  region  in  which 
the  function  is  holomorphic.  We  shall  now  show  that  had  we 
known  merely  the  v^alues  of  the  function  in  ever  so  small  a  neighbor- 
hood of  any  point  of  the  complex  plane  other  than  2=1,  say  the 
origin,  we  should  have  been  able,  at  least  theoretically,  to  com- 
pute the  value  of  the  function  at  every  point  of  the  region  in 
which  it  is  holomorphic  and  that  without  even  finding  the  expression 

:j at  all.     The  unique  determination  of  the  values  of  a  function 

in  a  more  or  less  extended  region  by  its  values  in  an  arbitrarily  small 
portion  of  that  region  is  a  general  property  of  functions  of  a  complex 
variable  for  regions  in  which  they  are  holomorphic.  This  property 
may  be  more  exactly  formulated  in  the  following  theorem. 


Art.  49.]  ANALYTIC  CONTINUATION  247 

Theorem  I.  If  a  Junction  f{z)  is  holomorphic  in  a  given  region  S, 
then  it  is  uniquely  determined  for  all  values  of  z  in  S  by  its  values  along 
any  arbitrarily  small  arc  of  an  ordinary  curve  proceeding  from  a  point 
ofS. 

Let  a  be  a  point  of  the  given  region  S  from  which  the  given  arc  is 
drawn  and  suppose  /3  to  be  any  other  point  of  S.  Let  a  and  /3  be 
connected  by  any  ordinary  curve  X  lying  wholly  within  S  and  coin- 
ciding with  the  given  arc  in  the  neighborhood  of  a.  Let  f{z)  be  holo- 
morphic in  S  and  suppose  its  values  to  be  given  along  that  portion  of 
X  lying  in  an  arbitrarily  small  neighborhood  of  a.  We  are  to  show 
that  /(/3)  is  then  uniquely  determined.  Since  f{z)  is  holomorphic  in 
the  neighborhood  of  a,  its  derived  functions  are  also  holomorphic  in 
the  same  neighborhood,  and  hence  for  z  =  a  the  successive  derivatives 
/'(a),  ria),   .   .   .   ,  /(")(«),   ...  (3) 

all  exist.     The  existence  of  the  derivative  /'(a)  implies  that  the  limit 

f(a-\-Az)-f(a) 
^     ~\ ' 

Az=0  AZ 

is  the  same,  when  Az  approaches  zero  in  any  manner  whatever. 
Hence,  the  value  of /'(a)  may  be  found  from  the  given  values  of /(z) 
by  taking  this  limit  as  z  approaches  a  along  any  curve  proceeding 
from  a,  for  example  along  the  given  curve  X .  The  higher  derivatives 
are  likewise  determined  by  the  given  values  of  /(z).  Knowing  the 
value  of  /(z)  for  z  =  a  and  the  successive  derivatives  given  in  (3) 
we  may  now  write  out  Taylor's  expansion  of  /(z)  for  values  of  z  in  the 
neighborhood  of  a,  namely, 

f"(a)  f  (")(«) 

/(a)+r(a)(z-a)+-?^(z-a)2+  •  •  •  ^^-^{z-aY+  •  •  •  .      (4) 

This  series  converges  and  defines  an  element  (/>o(z)  which  is  equal  to 
/(z)  for  all  values  of  z  within  any  circle  drawn  about  a  as  a  center 
and  lying  wholly  within  the  region  S.  Let  Co  be  a  circle  satisfying 
these  conditions.  If  the  point  /S  lies  within  Co,  then  the  value  of 
/(/3)  is  already  seen  to  be  uniquely  determined;  for,  in  order  to  find 
this  value  all  we  need  to  do  is  to  substitute  /3  for  z  in  series  (4). 
If  j3  lies  outside  of  Co,  let  a\  be  a  point  of  intersection  of  the  curve  X 
with  Co.  Take  a  point  Zi  on  the  given  curve  arbitrarily  close  to 
ai  but  within  Cq.  The  function  /(z)  is  holomorphic  in  the  neighbor- 
hood of  z  =  zi  and  the  successive  derivatives  of  /(z)  for  this  value  of 
2  can  be  found  by  successively  differentiating  (4)  term  by  term  and 


248 


SINGLE-VALUED  FUNCTIONS 


(Chap.  VII. 


substituting  Zi  for  z  in  the  several  derived  series.  The  coefficients  of 
Taylor's  expansion  of  f{z)  about  the  point  Zi  are  therefore  uniquely 
determined.     The  resulting  expansion  is 


/(^o+rc^oc^-^o+'^c^- 


■z,y-\- 


+ 


/(">(2i) 


{z-ziy-\- 


(5) 


This  series  in  turn  defines  an  element  <t>i{z)  which  is  identical  with 
/(«)  for  all  values  of  z  within  a  circle  Ci  drawn  about  Zi  as  a  center 

and  lying  wholly  in  S.  Since  zi  was 
taken  arbitrarily  close  to  ai  and  since 
ai  is  an  inner  point  of  S,  the  circle 
Ci  must  intersect  Co.  If  jS  lies  within 
the  circle  Ci,  the  value  of  /(jS)  is 
uniquely  determined;  for,  to  find/(i3) 
we  need  merely  to  replace  z  by  /3  in 
(5).  If  P  lies  outside  of  Ci,  then  take 
a  point  22  on  X  lying  arbitrarily  near 
the  point  02  where  X  cuts  Ci  and 
compute  as  before  the  Taylor's  ex- 
pansion of  f(z)  for  values  of  z  in  the 
neighborhood  of  the  point  z  =  z^. 
Proceeding  in  this  manner,  it  is  pos- 
sible, at  least  theoretically,  to  obtain 
after  a  finite  number  of  operations  a 
series  which  converges  within  a  circle 
Ck  lying  wholly  within  *S  and  having 
/3  as  an  inner  point.  By  substituting 
/3  for  2  in  this  series  the  value  of  /(/3)  can  be  found  and  hence  the 
given  function  is  uniquely  determined  for  2  =  /3.  However,  /3  is  any 
point  of  S  and  hence  the  given  function  f(z)  is  uniquely  determined 
for  all  points  of  S  as  stated  in  the  theorem. 

As  a  direct  consequence  of  the  foregoing  theorem,  we  have  the 
following  corollary. 

CoROLLAHY.  If  two  fuuctions  are  holomorphic  in  a  given  region  S 
and  are  equal  for  all  values  of  z  in  the  neighborhood  of  a  point  z  =  a  of 
S,  or  for  all  values  of  z  along  an  arbitrarily  small  arc  proceeding  from  a, 
then  the  two  functions  are  equal  for  all  values  of  z  in  S. 

Thus  far  we  have  discussed  for  the  most  part  functions  which 
were  known  to  be  holomorphic  in  a  given  region.  We  have  not  in- 
quired into  the  question  as  to  how  large  that  region  might  be  in  any 


Fig.  88. 


Art.  49.]  ANALYTIC  CONTINUATION  249 

given  case.  By  aid  of  the  foregoing  corollary  we  can  now  consider 
the  possibility  of  extending  the  region  in  which  a  function  is  known 
to  be  holomorphic.  Moreover,  we  shall  be  able  to  show  that  an 
analytic  function  is  completely  and  uniquely  determined  if  it  is 
known  for  any  region  however  small  that  region  may  be. 

Let  ^i(z)  be  defined  for  the  region  Si.  Fig.  89,  and  in  this  region 

0 


Fig.  89.  Fig.  90. 

let  it  be  holomorphic.  Suppose  it  is  possible  to  find  a  second  func- 
tion (hi^)  which  is  holomorphic  in  an  adjacent  region  S2,  having  an 
arc  C  of  an  ordinary  curve  for  at  least  a  portion  of  the  boundary 
between  Si  and  S2.  Moreover,  let  (l>i{z),  4>2{z)  be  each  defined  for 
values  of  z  along  C,  end  points  excepted,  and  for  these  values  let  each 
of  these  functions  be  holomorphic  and  equal  to  the  other.  Then  for 
values  of  z  in  Si,  S2  and  along  C  the  functions  0i(z),  <^(2)  define  a 
function  f{z)  which  is  holomorphic  in  this  enlarged  region.  The 
function  <^(z)  is  called  an  analytic  continuation  of  <^i(z),  and  the 
process  of  finding  such  a  function  is  called  the  process  of  analytic 
continuation. 

It  follows  from  the  corollary  to  Theorem  I  that  for  values  of  z  in 
jS2,  the  function  /(z)  thus  defined  is  uniquely  determined.  For  sup- 
pose that  another  analytic  continuation  of  <^i(z),  say  $i(z),  could  be 
found  such  that  in  the  region  ^2  it  has  values  different  from  02(z). 
We  should  have  a  function  F(z),  defined  by  <t>i(z)  and  $i(z),  which 
is  also  holomorphic  in  the  region  Si  -\-  S2  -\-  C,  that  is,  defined  for 
values  of  z  in  Si,  S2  and  along  the  arc  C.  We  then  have  two  functions 
/(z)  and  F{z)  each  holomorphic  in  the  region  S  =  Si  -\-  S2  -\-  C  and 


250  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

^ual  to  each  other  in  Su  By  the  foregoing  corollary  these  two  func- 
tions must  be  identical  throughout  the  region  S. 

It  is  evident  that  <l>i{z)  likewise  may  be  regarded  as  an  analytic 
continuation  of  <hi^)-  Either  of  these  functions  is  uniquely  deter- 
mined when  the  other  is  known.  Instead  of  having  merely  boundary 
points  in  common,  the  two  regions  ^i,  *S2  may  of  course  overlap.  At 
the  points  common  to  the  regions  Si,  S2  the  two  functions  01(2),  (^(2) 
must  in  this  case  satisfy  the  same  conditions  as  at  points  along  the 
arc  C  in  the  case  where  the  regions  are  adjacent  but  do  not  over- 
lap, namely,  they  must  have  equal  values  and  be  holomorphic.  In 
both  cases  we  speak  of  the  functions  ^1(2),  <^(2)  as  elements  of  the 
function /(«). 

If  an  element  <t)i{z)  of  a  function  is  holomorphic  in  the  neighbor- 
hood of  a  point,  we  say  that  0i(z)  is  continued  analytically  along  a 
curve  from  a  to  /3  if  this  curve  lies  wholly  within  a  finite  sequence  of 
connected  regions  Si,  S2,  .  .  .  ,  Sn  such  that  <t>i(z)  has  an  analytic 
continuation  (^(2)  defined  for  S2  and  <t>2{z)  likewise  can  be  continued 
analytically  in  the  region  S3,  etc. 

The  following  theorem  *  sets  forth  the  necessary  and  sufficient 
conditions  for  analytic  continuation. 

Theorem  II.  Given  two  functions  </>i(z),  ^(2)  which  are  holo- 
morphic respectively  in  the  adjacerd  regions  Si,  S2  having  an  arc  C  of 
an  ordinary  curve  as  that  portion  of  their  boundaries  common  to  the  two. 
The  necessary  and  sufficient  condition  that  each  of  these  functions  is 
an  analytic  continuation  of  the  other  is  that  they  converge  uniformly 
to  equal  values  on  C. 

It  follows  at  once  from  the  definition  of  analytic  continuation  that 
the  conditions  set  forth  in  the  theorem  are  necessary;  for,  if  either  is 
an  analytic  continuation  of  the  other,  then  they  must  have  equal 
values  along  C,  and  moreover,  for  these  values  they  must  be  holo- 
morphic and  hence  continuous  with  the  values  at  points  within  Si, 
Sz,  respectively. 

To  show  that  the  given  conditions  are  sufficient  we  define  /(z)  as 

follows: 

/(2)  =  <t>i{z)  in  ,Si 
=  <^(z)  in  Si 
=  ^1(2)  =  ^(2)  along  C. 

*  See  Painlev^,  Toulouse  Annates,  Vol.  II,  p.  28;  also  Pompeiu,  UEnseigne- 
ment  Malhemaiique,  July,  1913,  p.  305. 


Art.  49.] 


ANALYTIC  CONTINUATION 


251 


We  must  then  show  that  /(z)  is  holomorphic  in  the  region  >S  com- 
posed of  Si,  Si  and  the  points  of  C,  end  points  excepted. 

Let  7i,  72  be  two  ordinary  curves 
joining  any  two  points  A,  B  oi  C 
and  lying  wholly  within  the  regions 
Si,  S2,  respectively.  By  the  Cauchy- 
Goursat  theorem,  we  have 

/  (j>i{z)  dz  +    f(i>i{z)  dz  =  0,       (6) 
«/7i  Jab 

/  <h{z)  dz-\-   I  <h{z)  dz  =  0.       (7) 
t/72  *Jba 

Denote  by   7    the   curve  71  +  72. 

Combine  the  integrals  in  (6)  with 

the  corresponding  integrals  of  (7). 

Since  <^i(z)  =  ^2(2)  for  values  of  z  Fig.  91. 

along  C,  we  have 

/  01(2)  dz-\-   I  <j)2{z)  dz  =  0. 
Jab  Jba 

By  definition /(z)  is  equal  to  ^1(2)  for  values  of  z  in  Si  and  to  ^(z)  for 
values  of  z  in  S2.    Hence,  we  have  from  (6)  and  (7) 


Ifiz)  dz=    I  <l>i{z)  dz+    I  <h{z)  dz  =  0. 


(8) 


Since  7  is  any  ordinary  curve  lying  wholly  within  S  and  passing 
through  A,  B,  it  follows  from  Morera's  theorem  that  J{z)  is  holo- 
morphic for  all  values  of  2  in  5  and  hence  for  all  values  of  the  vari- 
able along  the  curve  C,  end  points  excepted.  Consequently,  either 
of  the  two  functions  ^i,  02  may  be  regarded  as  the  analytic  continu- 
ation of  the  other.  Moreover,  it  follows  from  our  previous  discus- 
sion that  either  of  these  functions  is  uniquely  determined  when  the 
other  is  known. 

•  One  of  the  methods  of  analytic  continuation  most  frequently  em- 
ployed in  theoretical  discussions  is  that  by  means  of  power  series. 
Suppose  we  have  given  a  power  series,  say  of  the  form 

ceo  +  ai  (2  -  Zo)  +  0:2(2  -  2o)2  +   •  •  •   +  an{z  -  2o)"  +   •  •  •  .       (9) 

Within  its  circle  of  convergence  Co  this  series  defines  an  element 
00(2)  which  is  holomorphic  within  Cq.  Let  Zi  be  any  point  within  Co 
but  lying  arbitrarily  close  to  Co.     Since  00(2)  is  holomorphic  in  the 


k 


252  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

neighborhood  of  2  =  Zi,  we  can  compute  Taylor's  expansion  of  ^(2), 
obtaining 

Uzi)  +  <t>o'(zO  {z  -  21)  +  ^^  {z  -  z,Y 

+    ...+^(,_,0n+..-.  (10) 

n  ! 

Let  Ci  be  the  circle  of  convergence  of  the  series  (10);  then  within 
Ci  this  series  defines  an  element  <t>i{z)  which  is  holomorphic  within 
C\.  Suppose  Ci  intersects  Co;  then  in  the  region  included  within  the 
two  circles  Co  and  C\,  the  elements  ^0(2),  0i(z)  satisfy  the  condi- 
tions of  analytic  continuation  and  consequently  define  a  function 
f{z)  which  is  holomorphic  in  this  region.  Either  of  these  elements 
may  be  again  continued  by  the  same  means.  By  this  process  any 
given  element  may,  at  least  theoretically,  be  continued  analytically 
by  the  method  of  power  series  until  the  whole  complex  plane,  with 
the  exception  of  certain  points  or  regions  excluded  by  the  inherent 
character  of  the  function  /(z)  so  defined,  is  covered  by  overiapping 
circles  of  convergence.  The  character  of  the  exceptional  points  and 
regions  will  be  discussed  in  subsequent  articles. 
^  The  importance  of  the  power  series  method  of  analytic  continu- 
ation is  due  largely  to  its  theoretical  value.  Other  methods,  although 
restricted  in  their  uses  for  theoretical  discussions,  are  of  much  greater 
practical  importance  in  the  applications  of  the  theory  of  functions. 
We  shall  now  consider  a  method  introduced  by  Schwarz,*  in  which 
he  makes  use  of  the  principle  of  symmetry.  Let  ^1(2)  be  a  function 
which  is  holomorphic  in  a  region  Si  lying  in  the  upper  half-plane  and 
having  a  segment  AB  oi  the  axis  of  reals  as  a  part  of  its  boundary. 
Suppose  that  as  2  approaches  any  point  x  oi  AB  along  any  path 
whatsoever  lying  interior  to  *Si,  <i>i{z)  approaches  a  definite  real  value 
^i(x).  Then  by  Art,  13  <^i(x)  is  a  continuous  function  of  x.  Denote 
by  z  a  point  in  the  lower  half-plane  situated  symmetrically  with  re- 
spect to  2  relative  to  the  axis  of  reals.  The  assemblage  of  points  2 
constitutes  a  region  ^2  symmetrical  to  >Si  with  respect  to  AB. 
Associate  with  each  value  of  2  a  functional  value  which  is  the  con- 
jugate imaginary  of  <t>i{z).  The  assemblage  of  these  values  defines  a 
function  <i>i{z)  which  is  holomorphic  in  82  and  converges  to  the  real 
values  <h{x)  =  <f>i(x)  along  the  axis  of  reals. 

*  See  Crelle,  Vol.  LXX,  pp.   106,   107;    also  MathemcUische  Abhandlungen, 
Vol.  II,  pp.  65-83. 


Art.  49.] 


ANALYTIC    CONTINUATION 


253 


In  the  continuous  region  *S  made  up  of  Si,  S2  and  the  points  along 
the  axis  of  reals  between  A  and  B,  the  functions  ^i(z),  <h{z)  satisfy 
the  conditions  of  Theorem  II  and  hence  (^(0)  is  an  analytic  continu- 
ation of  0i(0).  Each  of  these  func- 
tions is  then  an  element  of  a  func- 
tion f(z)  which  is  holomorphic  in 
S  and  is  equal  to  <t>i(z)  in  Si  and 
equal  to  <^(z)  in  S2,  and  moreover 
f{z)  takes  the  common  values  of 
the  elements  <f)i{z),  <h{z)  along  the 
axis  of  reals  between  A  and  B. 
The  advantage  of  this  method  of 
analytic  continuation  is  the  ease 
with  which  the  continuation  can 
be  obtained.  All  that  is  needed  is 
to  reflect  the  given  region  upon 
the  X-axis  and  associate  with  the  reflected  region  a  function  which 
is  the  conjugate  imaginary  function  of  0i(z). 

We  shall  now  consider  a  generalization  of  the  foregoing  method  of 
analytic  continuation.  To  do  so  we  shall  make  use  of  a  generaliza- 
tion of  the  idea  of  reflection.  Let  the  points  of  the  segment  AB  oi 
the  axis  of  reals,  which  formed  a  common  portion  of  the  boundary 
between  the  two  given  regions,  be  made  to  correspond  to  the  points 
of  a  regular  arc  C  of  an  analytic  curve.  By  an  analytic  curve  is 
understood  one  whose  parametric  equations  are  of  the  form 

X  =  ^:(0,       y  =  ^2(0, 


Fig.  92. 


(11) 


where  ^i(0,  ^2(0  are  real,  analytic  functions  of  the  real  variable  L 
An  arc  of  such  a  curve  is  regular  if  we  have  the  added  condition  that 
the  derivatives  ^i'(0,  ^2'(0  are  not  simultaneously  zero;  that  is,  if 
we  have 

To  any  point  U  oi  AB  there  corresponds  a  point  Zq  =  (xo,  yo)  of  C. 
As  the  two  functions ^1(0, "^2(0  are  analytic,  each  may  be  expanded  in 
powers  of  {t—tn).  The  resulting  series  converge  for  all  values  of  the 
variable  within  their  circles  of  convergence,  and  hence  t  may  take 
complex  as  well  as  real  values.  Denoting  these  real  and  complex 
values  by  t,  we  have 

z  =  x  +  iy  =  ^i(t)  +  i^2(r)  =  <^{r),  (12) 


254 


SINGLE-VALUED  FUNCTIONS 


[Chap.  VII. 


which  is  holomorphic  and  has  a  derivative  different  from  zero  for  all 
points  of  AB,  end  points  at  most  excepted.  The  function  z  =  "^(r) 
is  then  defined  for  a  region  S  of  the  r-plane  consisting  of  the  inner 
points  oi  AB  and  certain  regions  *Si,  S2.  lying  sjamngijcically  with 
respect  to  AB,  Fig.  93.  By  Theorem  II,  Art.  21,  there  exists  in  the  Z- 
plane  a  corresponding  region  *S'  consisting  of  the  points  of  C  and  the 
regions  Si,  8%  lying  on  either  side  of  C,  in  which  the  inverse  function 
T  =  <t>{z)  is  uniquely  determined  and  holomorphic.  The  region  S  can 
be  so  restricted  that  the  function  z  =  ^(t)  and  its  inverse  function 
T  =  <i>{z)  map  the  regions  S,  S'  upon  each  other. 


Fig.  93. 

Then  to  any  conjugate  imaginary  points  ti  and  n  lying  respec- 
tively in  Si  and  S2,  there  are  associated  two  corresponding  2-points 
namely  Zi  and  Zi  lying  respectively  in  Si  and  *S2',  and  conversely. 
It  is  to  be  noted  that  the  particular  values  of  z  thus  associated  depend 
upon  the  form  of  the  curve  C  and  not  upon  the  form  of  the  parametric 
equations  (11)  of  the  curve.  Suppose,  for  example,  a  different  para- 
metric representation  of  the  curve  C  is  obtained  by  replacing  t  in  (11) 
by  an  analytic  function  of  any  other  real  variable  r.  If  we  then 
permit  r  to  take  complex  values,  conjugate  imaginary  points  in  the 
T-plane  correspond  to  conjugate  imaginary  points  in  the  r-plane,  and 
consequently  we  get  the  same  corresponding  values  of  z. 

Of  the  two  z-points  corresponding  to  conjugate  imaginary  values 
of  a  parameter  r,  either  is  said  to  be  the  reflection  or  image  of  the 
other  with  respect  to  the  curve  C.  Likewise  the  region  ^2'  may  be 
spoken  of  as  the  reflection  of  the  region  Si  with  respect  to  C. 

This  definition  of  reflection  with  respect  to  a  regular  arc  of  an 
analytic  curve  may  now  be  used  in  developing  a  method  of  analytic 
continuation.  Let  Si,  S2  be  any  two  adjacent  regions  such  that  S2 
is  the  reflection  of  Si  with  respect  to  the  regular  arc  C  of  an  analytic 
curve  whose  parametric  equations  are  x  =  '^i{t),  y  =  ^2(0-  Let 
<f>i{z)  be  a  function  which  is  holomorphic  in  Si  and  defined  for  values 


Art.  49.]  ANALYTIC  CONTINUATION  255 

of  z  along  C  by  its  limiting  values  as  z  approaches  the  points  of  C 
by  any  path  whatever  lying  wholly  within  *Si'.  We  can  now  state  in 
the  following  form  the  necessary  and  sufficient  condition  that  ^\{z) 
may  be  analytically  continued  by  reflection  with  respect  to  arc  C. 

Theorem  III.  The  necessary  and  sufficient  condition  that  <})i{z) 
can  be  analytically  continued  by  reflection  with  respect  to  the  regular 
arc  C  of  an  analytic  curve  forming  a  portion  of  the  boundary  of  the 
region  for  which  <f>i(z)  is  defined  is  that  <f)i{z)  converges  uniformly  to 
real  values  along  C. 

If  4>i{z)  can  be  analytically  continued  across  the  arc  C  into  the 
region  *S2',  which  is  a  reflection  of  >Si'  with  respect  to  C,  then  for  the 
region  AS2'  a  function  <h.{z)  is  determined  such  that  (i>\{z),  <fyi{z)  define 
a  function /(z),  holomorphic  in  the  region  ;S'  consisting  of  the  points 
of  C  and  the  regions  Si,  S2'.  Along  C  the  functions /(z),  0i(z),  <t>2{z) 
take  equal  values  which  are  continuous  with  the  values  taken  respec- 
tively in  the  regions  ^1',  82^    As  may  be  seen,  the  substitution 

z  =  X  -\-  iy  =  ^(t) 

transforms  the  functions  <^i(z),  <h{^)  into  the  functions  Fi{t),  Fzir) 
which  are  holomorphic  in  Si,  S2,  respectively,  and  along  AB  take  equal 
values.  Moreover,  since  ^2'  is  a  reflection  of  Si  with  respect  to  C, 
S2  is  likewise  a  reflection  of  *Si  with  respect  to  AB.  The  function /(z) 
is  likewise  transformed  into  a  function  F{t),  holomorphic  in  the  region 
S  consisting  of  the  points  of  AB  and  the  regions  Si,  S2,  such  that  it 
coincides  with  Fi{t)  in  ^1  and  with  F^ir)  in  *S2  and  along  ABwe  have 

Fit)  =  Flit)  =  F^it). 

The  function  F2(t)  is  therefore  an  analytic  continuation  of  Fiir) 
by  Theorem  II. 

In  a  similar  manner  the  substitution 

T   =   <i>iz) 

transforms  the  functions  Fi(t),  Fiir),  Fir)  into  the  functions  <f>iiz), 
'(fniz),  fiz),  respectively,  where,  if  ^2(7)  is  an  analytic  continuation  of 
Fi(t),  then  02(2)  is  likewise  an  analytic  continuation  of  ^1(2),  such 
that  0i(z),  ^(z)  take  equal  values  with/(z)  for  values  of  z  along  C. 

Consequently,  we  see  that  whenever  ^2(2)  is  an  analytic  continu- 
ation of  01  (z),  then  F^ir)  is  an  analytic  continuation  of  Fi(t)  and  con- 
versely. The  necessary  and  sufficient  condition  that  F^ir)  is  an 
analytic  continuation  of  i^i(r)  leads  then  to  the  necessary  and  sufficient 


256 


SINGLE-VALUED  FUNCTIONS 


[Chap.  VII. 


condition  that  <^(2)  is  an  analytic  continuation  of  4>i{z).  Moreover, 
if  Fj(t)  is  obtained  as  an  analytic  continuation  of  Fi(t)  by  means  of 
Schwarz's  method  of  reflection,  then  ^(2)  is  a  continuation  of  <^i(2) 
by  reflection  with  respect  to  the  arc  C.  But  as  we  have  seen  the 
necessary  and  sufficient  condition  that  Fi{t)  can  be  analytically 
continued  by  reflection  upon  AB  is  that  Fi(t)  takes  along  AB  real 
values  which  are  continuous  with  the  values  taken  by  this  function 
in  Si;  that  is,  that  Fi{t)  converges  uniformly  toward  real  values 
along  AB.    Accordingly  the  necessary  and  sufficient  condition  that 

<f>i(z)  can  be  analytically  continued 
across  C  by  reflection  is  that  <f>i{z) 
converges  imiformly  to  real  values 
along  the  arc  C  as  the  theorem  re- 
quires. 

Let  us  now  apply  this  method  of  analytic 
continuation  by  reflecting  a  given  region 
with  respect  to  an  arc  of  a  circle.     Let  C 
be  any  circle  having  its  center  at  0,  Fig.  94. 
Let  the  element  4>i(z)  of  the  function  /(z) 
be  defined  for  the  region  S  bounded  by 
three  arcs  Ci,  C2,  C3  of  circles  cutting  the 
circle  C  at  right  angles  and  suppose  that 
<t>i{z)    converges   uniformly  to  real  values 
along  Ci,  C2,  C3. 
We  shall  now  reflect  the  region  S  with  respect  to  one  of  these  arcs,  say  the  arc 
Ci.     In  order  to  accomplish  this  we  shall  first  show  that  the  reflection  of  any 
point  of  S  with  respect  to  Ci  is  the  conjugate  point  obtained  by  geometric  inver- 
sion of  the  given  point  with  respect  to  the  circle  of  which  Ci  is  an  arc. 

To  show  this  consider  the  parametric  equations  of  Ci.     Suppose  the  center  of 
Ci  to  be  the  origin  and  its  radius  to  be  unity.     Then  any  functions 

X  =  *i(0,      y  =  *2(0, 

which  satisfy  the  equation 

a;2  4.  y2   =   1 

will  answer  our  purpose.     For  example,  we  may  put 

l-C^  2t 


1  +^»' 


y  = 


1+^ 


(13) 


The  functional  relation  (12)  which  maps  ther-plane  upon  the  Z-plane  is  then 

1  -T«    .    .      2i 


z  =  X  -\-iy  = 


+  i 


1  -f-T*    '       l+r* 
l+2ir-T»^  (l+tr)» 
1  +  T*  l-h  r» 

1+tT 

l-ir' 


(14) 


Art.  50.]  ANALYTIC   FUNCTIONS  257 

Let  Zi,  Zi  be  the  points  of  the  Z-plane  corresponding  respectively  to  the  conjugate 
imaginary  values  n  +  vti,  n  —  ir^  of  the  r-plane.    We  have  then  from  (14) 

.  1  +  i{rx  +  trz)        1-firi-Tj 

Z\=  X\-\-  lyi  =  :} -, r— T-T  =   :; : j > 

1    —  1(ti  +  iTi)  1   -  ITI   +  T2 

and  22  =  X2  +  17/2  =  :j V ^~T  =  Ti '■ ' 

1    —  l(jl    —  iTl)  1    —   iTl    —  T% 

whence,  we  obtain 

Xz  +  iyi  =  z r-^.  (15) 

Xi  —  lyi 

,    .         Xi  -\-  iyi 
or  X2  +  iy2=     ol    ,• 

xi^  +  yi* 

Equating  the  real  parts  and  the  imaginary  parts,  we  have 

^  =  ^2!  2'     y^  =   2?  2-  (16) 

By  comparing  with  the  equations  of  transformation  given  in  Art.  38,  it  will  be 
seen  that  the  reflection  of  zi  with  respect  to  the  arc  Ci  is  merely  the  conjugate 
point  of  2i  with  respect  to  Ci. 

The  foregoing  conclusion  gives  us  a  convenient  method  for  determining  the 
region  Si  which  is  the  reflection  of  S  with  respect  to  Ci  and  hence  for  determining 
an  analytic  continuation  of  0i(z).  Since  C  cuts  Ci  at  right  angles,  the  circle  C 
inverts  into  itself.  The  points  a,  b  remain  imchanged,  and  the  point  c  goes  over 
into  c'.  As  C2  likewise  cuts  the  circle  C  at  right  angles,  it  inverts  into  a  circle 
C2'  perpendicular  to  C  and  passing  through  c'  and  b.  In  a  similar  manner  the  curve 
C3  goes  over  into  the  curve  d'  passing  through  the  points  a,  c'  and  cutting  the 
circle  C  at  right  angles.  The  region  S  is  then  reflected  into  the  region  Si.  Asso- 
ciating with  each  point  of  Si  the  conjugate  imaginary  value  of  <l>i{z),  where  z  is 
the  corresponding  point  in  S,  we  have  by  Theorem  II  </>2(z),  an  analytic  continu- 
ation of  01  (z). 

In  a  similar  manner  ^3  (2)  is  an  analytic  continuation  of  the  given  element 
<f>i(z)  by  reflection  with  respect  to  d  and  04  (z),  by  reflection  with  respect  to  Cs. 
Continuing  this  process  it  is  possible  to  enlarge  the  region  S  originally  given,  so 
as  to  include  in  the  limit  the  entire  region  bounded  by  C. 

50.  Analytic  function.  By  the  aid  of  the  results  of  the  preced- 
ing article  concerning  analytic  continuation,  we  can  formulate  more 
exactly  the  definition  of  an  analytic  function.  If  we  know  the  values 
of  a  function  and  its  derivatives  at  any  point  a,  then,  as  we  have 
already  seen,  an  element  ^1(2)  of  that  function  is  uniquely  deter- 
mined. By  analytic  continuation  we  can  extend  the  region  in  which 
the  function  is  thus  defined  by  determining  other  elements  of  the 
function  and  their  corresponding  regions.  This  extended  region 
forms  a  connected  region  S  within  which  a  function  is  defined  by 
means  of  its  elements.  If  we  now  suppose  the  region  S  to  be  extended 
as  far  as  possible  by  means  of  analytic  continuation,  then  the  corre- 


258  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

sponding  aggregate  of  elements  fully  defines  a  function  f{z)  in  S 
such  that  /(z)  is  equal  to  each  of  its  elements  (/>(z)  for  those  values  of  z 
for  which  4>{z)  is  defined.  The  function  f{z)  so  defined  is  called  a  mono- 
genic analjrtic  fxinction,  or  more  briefly  an  analytic  function.  As  it 
is  impossible  to  further  extend  this  region  S,  it  is  called  the  region  of 
existence  of  the  analytic  f unction /( 2) .  The  element  from  which  the 
other  elements  are  obtained  by  the  process  of  analytic  continuation 
is  called  the  primitive  element  of  the  function,  and  the  remaining 
elements  become  analytic  continuations  of  it. 

The  region  of  existence  consists  of  a  continuum  of  inner  points, 
each  of  which  is  a  regular  point  of  the  function /(z).  The  region  of 
existence  may  extend  over  the  entire  finite  portion  of  the  complex 
plane.  On  the  other  hand,  it  is  possible  in  the  process  of  analytic 
continuation  to  encounter  a  closed  curve  beyond  which  the  function 
can  not  be  analytically  continued.  In  such  a  case  the  curve  is  called 
a  natural  boundary.  For  example,  in  the  function  discussed  in  con- 
nection with  Fig.  94  of  the  last  article,  the  curve  C  constitutes  a 
natural  boundary,  since  it  is  impossible  to  continue.--Uie__£unction 
analytically  across  this  curve.  A  portiwi  of  the  complex  plane  into 
which  the  function  can  not  be  continued  because  of  a  natural  boundary 
is  called  a  lacunary  space.  Often,  instead  of  a  lacunary  space,  we 
encounter  a  set  of  points,  not  constituting  a  continuum,  which  can 
not  be  included  in  the  region  of  existence.  Such  points  are  not 
regular  points  of  the  function,  and  hence  they  must  be  classed  as 
singular  points.  The  various  classes  of  singularities  of  single-valued 
analytic  functions  will  be  more  fully  discussed  in  the  following  article. 

Since  power  series  may  be  used  as  a  means  of  analytic  continuation, 
it  follows  that  an  analytic  function  may  also  be  defined  as  one  that 
is  developable,  except  in  the  neighborhood  of  singular  points,  by 
Taylor's  expansion.  It  is  to  be  noted  also  that  a  single-valued 
analytic  function  of  a  complex  variable  is  uniquely  determined 
throughout  its  region  of  existence  as  soon  as  its  values  in  the  neigh- 
borhood of  any  regular  point  of  that  region  are  given.  The  particular 
method  of  analytic  continuation  employed  in  extending  the  region 
from  the  neighborhood  of  the  given  point  to  the  region  of  existence  of 
the  function  thus  determined  is  a  matter  of  indifference.  Moreover, 
any  two  analytic  functions  are  equal  for  all  values  of  z  in  this  region 
of  existence  if  they  have  a  common  element. 

An  important  distinction  between  functions  of  a  complex  variable 
and  those  of  a  real  variable  may  be  noted.     If  a  function  of  a  com- 


Art.  50.]  ANALYTIC   FUNCTIONS  259 

plex  variable  has  a  derivative  at  each  point  of  a  given  region  S,  then 
at  all  points  of  S  it  has  derivatives  of  every  order,  and  in  the  neigh- 
borhood of  any  point  of  S  the  function  is  represented  by  the  Taylor 
series  to  which  it  gives  rise.  On  the  other  hand,  it  does  not  follow 
that  if  a  function  of  a  real  variable  has  at  each  point  of  an  interval 
derivatives  of  every  order  that  the  Taylor's  expansion  derived  from 
the  function  represents  that  function  for  all  values  of  the  variable  in 
the  neighborhood  of  the  point  at  which  the  derivatives  are  taken. 
The  following  example  will  serve  as  an  illustration. 

Ex.  1.     Given  the  function  f{x)  =  e    x^,  x  ^  0,  where  /(O)  =  0.     Let  it  be 
required  to  determine  the  interval  of  equivalence  of  this  function  and  the  power 
series  obtained  from  this  function  by  expanding  it  in  a  Maclaurin  series. 
We  have,  for  x  5^  0, 

_^ 
fix)  =e    -\ 

-\         e-h 
_i  _i 


where  G'(x)  is  a  polynomial  in  x. 

For  X  =  0  we  have  by  use  of  the  limit  * 

Az=0  ^  Ai=0      ^^ 

forr,.   _    r       /(AX)  -f  (0)   _  eZ^  _ 

^  ^^^-^.0  Ax  -  Jio^    (Ax)*    -0' 

,.,0)=    L    r(A^)-r(0)        ,J4-6(AX)^|^  =  0, 


Ai=0 


Ax  Az=0  (Ax)^ 


*  See  Stolz,  Differenticd-und  Integralrechnung,  p.  76,  also  p.  81,  Ex.  3. 


260 


SINGLE-VALUED  FUNCTIONS 


[Chap.  VII. 


The  expansion  derived  from  the  given  function  is  a  power  series,  each 
term  of  which  is  zero.  This  series  defines,  in  the  interval  of  convergence,  a  func- 
tion 4>(x)  =0;  that  is,  the  function  defined  by  the  series  is  represented  geometric- 
ally by  the  X-axis.  On  the  other  hand,  the  given  function  y  =  fix)  is  represented 
by  the  curve  C  tangent  to  the  X-axis  at  the  origin.     This  curve  is  symmetrical 


Fig.  95. 


-7 


with  respect  to  the  F-axis,  and  has  the  line  y  =  1  as  an  asymptote  as  shown  in 
Fig.  95.  It  follows  that  the  function  that  gave  rise  to  the  series  is  represented 
by  that  series  in  only  one  point,  namely  x  =  0. 

The  reason  for  the  distinction  pointed  out  between  functions  of  a 

complex  variable  and  those  of  a  real  variable  is,  so  far  as  the  partic- 

_L 
ular  function  discussed  is  concerned,  that  while  e   *'  is  an  analytic 

function,  the  point  z  =  0  is  not  a  regular  point,  since  the  derivative 

with  respect  to  the  complex  variable  z  does  not  exist  at  the  origin; 

although  in  the  realm  of  real  variables  the  corresponding  derivative 

does  exist.    Consequently,  the  point  z  =  0  does  not  belong  to  the 

region  of  existence  in  the  complex  plane. 

It  is  of  importance  to  point  out  in  this  connection  the  distinction 

between  an  analytic  function  as  defined  and  an  analytic  expression. 

The  notion  of  an  analytic  function  implies  a  definite  correspondence 

between  the  z-points  and  the  ty-points  of  the  complex  plane.     This 

relation  may  have  different  forms  of  expression  in  different  parts  of 

the  plane.     An  analytic  expression  on  the  other  hand  is  the  result 

obtained  by  performing  upon  the  independent  variable  the  analytic 


Art.  50.]  ANALYTIC  FUNCTIONS  261 

operations  of  addition,  subtraction,  multiplication,  division,  inte- 
gration, etc.,  including  the  general  process  of  taking  the  limit.  It 
leads  to  a  formal  expression  of  the  relation  between  z  and  w.  This 
analytic  expression,  however,  may  define  for  different  regions  of  the 
plane  elements  of  different  analytic  functions.  The  following  illus- 
trations will  make  clear  the  distinction. 
Consider  the  analytic  expression 

This  series  converges  *  for  all  values  of  z  except  for  values  upon 
the  unit  circle  about  the  origin.  Within  this  circle  the  series  con- 
verges to  the  limit 

z 


/i(2)  = 

'alues  of  z  exterior  to 
limit 


1  -z 
For  values  of  z  exterior  to  the  unit  circle  the  series  converges  to  the 


/2(2)    = 


1-2 

Hence,  for  \z\  <  1,  E{z)  may  be  considered  as  an  element  of  the 

z 
analytic  function  :j ,  and  for  |  2  |  >  1  it  may  be  considered  as 

an  element  of  the  analytic  function  :j .  In  either  case  the  ele- 
ment Ei^z)  may  be  analytically  continued  over  the  entire  finite  plane 
with  the  exception  of  the  point  z  =  1,  but  in  the  one  case  the  result- 

z            .     .                      .    .        1 
ing  analytic  function  is  -i ,  while  in  the  other  it  is  :; 

^  ^  1  —  2'  1-2 

As  another  illustration,  suppose  we  have  two  detached  regions  ;Si, 
S)z.  Let  «^i(2),  ^i{z)  be  elements  of  two  distinct  analytic  functions 
/i(z),  ji{z).  Suppose  «^i(2)  to  be  defined  for  <Si,  within  which  it  is 
holomorphic,  and  along  the  boundary  Ci  of  >Si  let  it  converge  uni- 
formly. Let  <^(z)  be  defined  in  a  similar  manner  for  ^2  and  along 
its  boundary  C2.  Consider  the  analytic  expression 
1      C  <^i(0  dt        1     C  <h(t)  dt 

It  follows  from  Cauchy's  integral  formula  that  for  values  of  2  within 

Si  we  have 

lit)  dt 


2 
*  See  Bromwich,  Theory  of  Infinite  Series,  p.  254,  Ex.  4. 


262  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

<f)Jt) 
For  values  of  z  exterior  to  Si,  the  integrand     _    ,  considered  as  a 

function  of  t,  is  holomorphic  in  Si  and  hence  by  the  Cauchy-Goursat 
theorem  this  integral  vanishes.  Similarly,  the  second  integral  in  (1) 
defines  the  element  02  (2)  for  values  of  z  within  Sz  and  vanishes  for  all 
values  of  2  exterior  to  that  region.  Hence,  the  expression  E{z)  is 
equal  to  ^1(2)  for  values  of  2  within  Ci  and  to  (^2(2)  for  values  of  z 
within  C2.  It  follows  then  that  E{z)  defines  an  element  of  the  analytic 
function /i (2)  ovf^iz)  according  as  2  lies  within  Ci  or  C2. 

51.  Singular  points  and  zero  points.  We  have  defined  a  sin- 
gular point  of  a  function  (Art.  14)  as  a  point  that  is  not  a  regular 
point  of  the  function,  but  in  every  deleted  neighborhood  of  which 
there  are  regular  points.  As  we  have  seen  in  the  previous  article 
the  singular  points  of  an  analytic  function  are  to  be  considered  as 
boundary  points  of  its  region  of  existence,  and  they  may  even  form 
a  closed  curve  constituting  a  natural  boundary  of  such  a  function. 
K  a  is  a  singular  point  of  an  analytic  function  j{z),  then  either  the 
function  has  no  derivative  at  the  point  a  itself,  or  there  are  points 
in  every  neighborhood  of  a  at  which  the  function  has  no  derivative. 
In  either  case  the  higher  derivatives  of  the  function  can  not  exist 
at  a,  and  hence  the  function  does  not  permit  of  an  integral  power 
series  development  in  the  neighborhood  of  a.  If  we  undertake  by 
means  of  power  series  to  continue  analytically  an  element  of  an 
analjrtic  function  along  an  ordinary  curve  passing  through  a  singular 
point,  the  circles  of  convergence  within  which  the  successive  elements 
are  defined  grow  gradually  smaller  as  their  centers  approach  the  sin- 
gular point;  for,  as  the  function  is  always  holomorphic  within  these 
circles  none  of  them  can  ever  inclose  the  singular  point  itself. 

The  singular  points  of  a  single-valued  analytic  function  may  be 
classified  as  poles,  or  non-essential  singular  points,  and  essential 
singular  points.  The  point  2  =  a  is  a  pole,  or  non-essential  singu- 
lar point,  of  the  analytic  function  j{z)  if  there  exists  a  positive  inte- 
gral value  of  k  such  that  the  product 

(2  -  aYf{z) 

is  holomorphic  in  the  neighborhood  of  a  and  different  from  zero  for 
z  =  a.    The  integer  k  is  called  the  order  of  the  pole. 
Thus  the  point  2  =  2  is  a  pole  of  order  2  of  the  function 

^^^^       (2  -  2)2 


Art.  51.]  SINGULAR   POINTS,   ZERO   POINTS  263 

for,  multiplying  J{z)  by  (z  —  2)^  we  obtain  the  function  3  z^  +  1, 
which  has  the  point  z  =  2  as  a  regular  point  and  is  different  from 
zero  for  z  =  2.  If  A;  is  equal  to  one,  the  pole  is  often  referred  to  as  a 
simple  pole. 

If  no  finite  value  of  k  exists  such  that  the  singularity  of  the  single- 
valued  analytic  function  /(z)  at  a  point  a  is  removed  by  multiplying 
by  the  factor  (z  —  aY,  then  a  is  said  to  be  an  essential  singular 
point  of /(z).  If  a  singular  point  can  be  inclosed  in  a  circle,  however 
small,  having  that  point  as  a  center  and  containing  no  other  singular 
point  of  the  given  function,  then  the  point  is  said  to  be  an  isolated 
singular  point.  An  isolated  essential  singular  point  is  one  that  may 
be  inclosed  in  a  circle  containing  no  other  essential  singular  point. 
It  may,  however,  have  an  infinite  number  of  poles  in  its  neighborhood, 
as  we  shall  see  later.  A  point  may,  therefore,  be  an  isolated  essential 
singular  point  without  being  an  isolated  singular  point. 

The  following  theorem  due  to  Riemann  is  important  in  establish- 
ing the  character  of  a  function  at  a  point  in  the  deleted  neighborhood 
of  which  it  is  limited  in  absolute  value  and  holomorphic* 

Theorem  I.  Let  /(z)  he  holomorphic  in  a  given  region  S  except  at  the 
point  z  =  a,  where  the  behavior  of  the  function  is  not  known.  If  for  all 
values  of  z  9^  a  in  S  we  have 

\m\<M, 

where  M  is  some  finite  positive  number,  then  f{z)  approaches  a  definite 
limit  A  as  z  approaches  a  and  z  =  ais  a  regular  point  of  f{z)  or  may  be 
made  so  by  assigning  tof{z)  at  the  point  a  the  value  f  (a)  =  A. 

Denote  by  C  any  ordinary  curve  lying  wholly  within  S  and  inclos- 
ing only  points  of  S,  including 
the  point  a.  Let  z  be  any  point 
within  C  other  than  a.  About  a 
as  a  center  draw  a  circle  y  lying 
wholly  within  C  and  having  an 
arbitrarily  small  radius  p.  The 
radius  p  can  then  be  so  chosen 
that  z  lies  exterior  to  y.    We  have  then  from  Theorem  I,  Art.  20, 

1    rf{t)dt     1    rf{t)dt  . 

^^'^  -  2Vi  JcJ^  +  2Vi  JyT^'  ^^^ 

*  See  Osgood,  Bulletin  of  Amer.  Math.  Soc,  June  1896,  p.  298;  also  Lehrbuch 
der  Funktionentheorie,  Zweite  Auflage,  p.  310. 


264  SINGLE-VALUED   FUNCTIONS  [Chap.  VII. 

where  the  integral  in  each  case  is  taken  positively  with  reference  to 
the  region  interior  to  C  and  exterior  to  y.  Since  we  have  for  values 
of  t  upon  7 

|/(i)|<M,  |i-2|E|z-„|_p, 

\2TnJyt  —  z        2-17  {\z  —  a\  —  p)  Jy^      '       \z  —  a\—p 

for,    I   \dl\  =  2rrp\s  the  length  of  the  circle  7.     As  Af  and  |  2;  —  a  | 

are  finite,  the  value  of  this  integral  is  arbitrarily  small  for  sufficiently 
small  values  of  p.  As  this  integral,  however,  does  not  vary  with  p, 
it  must  be  equal  to  zero.     Consequently,  we  have  from  (1) 

ZmJc  t  —  z 

which  holds  for  any  point  zin  S  other  than  the  point  a. 

By  Theorem  III,  Art.  20,  however,  this  integral  defines  a  function 
F{z),  which  is  holomorphic  everywhere  within  the  region  bounded 
by  C,  including  the  point  z  =  a  itself.  The  function  F(z)  thus  de- 
fined coincides  with  the  given  function  f{z)  for  all  values  oi  z  9^  a. 
Since  F{z)  is  holomorphic  in  S,  we  have 

L  F(z)  =  F{a)  =  A. 

If  we  now  put 

then  F(z)  is  identical  with  f(z)  for  all  values  of  z  within  C  and  conse- 
quently 2  =  a  is  a  regular  point  of  the  given  function  and 

Lf{z)=f(cc). 

It  follows  as  a  result  of  this  theorem  that  all  isolated  finite  discon- 
tinuities of  an  analytic  function  may  be  removed  by  the  proper 
definition  of  the  function  at  the  critical  points.  Consequently,  we 
need  not  concern  ourselves  with  the  consideration  of  such  discon- 
tinuities. This  theorem  also  brings  out  an  important  distinction 
between  functions  of  a  complex  variable  and  real  functions  of  a  real 
variable,  which  is  best  illustrated  by  an  example. 

Given  the  real  function  j{x)  of  a  real  variable  x  defined  by  the 
relations 

f{x)  =  X  sin  - ,  for  a;  7^  0 
=  0,        fora;  =  0. 


Art.  51.]  SINGULAR  POINTS,   ZERO  POINTS  265 

This  function  is  continuous  at  the  origin,  but  has  no  derivative 
at  that  point  although  it  possesses  a  derivative  at  every  point  in  the 
neighborhood  of  the  origin.* 

A  function  j{z)  of  a  complex  variable  differs  from  a  function  of  a 
real  variable  in  that  if  /(z)  is  holomorphic  in  the  deleted  neighbor- 
hood of  any  point  a  and  continuous  at  a,  then  a  is  necessarily  a  regular 
point  of  the  given  function.  In  other  words,  a  single-valued  analytic 
function  can  not  fail  to  have  a  derivative  at  an  isolated  point  in  the 
neighborhood  of  which  it  is  continuous. 

From  Theorem  II,  Art.  49,  it  follows  that  if  f{z)  is  continuous  in  a 
given  region  S  and  if  every  point  of  S,  with  the  exception  of  the 
points  of  an  ordinary  curve  lying  wholly  within  *S,  is  a  regular  point 
of  J{z),  then  f{z)  is  holomorphic  in  S.  In  other  words,  an  analytic 
function  of  a  complex  variable  can  not  have  an  isolated  line  of  singu- 
lar points  in  a  region  S  in  which  it  is  continuous  and,  except  for  the 
points  of  this  line,  holomorphic.  If  an  analytic  function  has  all  of 
the  points  of  a  curve  as  singular  points,  then  that  curve  forms  a 
portion  of  the  boundary  of  the  region  of  existence.  If  there  exists  a 
closed  curve,  every  point  of  which  is  a  singular  point,  then  that  curve 
constitutes  a  natural  boundary  of  the  function;  for,  in  such  a  case 
the  function  can  not  be  analytically  continued  beyond  the  curve. 

We  may  state  the  following  theorem  concerning  an  analytic  func- 
tion. 

Theorem  II.     If  an  analytic  function  f{z)  yMcffis  not  identically 
zero  and  is  holomorphic  in  the  deleted  neighborhood  of  z  =  Zo,  then  if 
L  f(z)  =  0  we  can  write  f{z)  in  the  form 

iz  -  zoy<f>{z), 

where  k  is  a  positive  integer  and  <t>{z)  is  different  from  zero  for  z  =  z^ 
and  ha^  this  point  as  a  regular  point. 

It  follows  from  Theorem  I  that  f(z)  is  holomorphic  in  the  neighbor- 
hood of  Zo  and  for  Zq  we  have  /(zo)  =  0.  We  may  then  expand  /(z) 
in  powers  of  (z  —  Zo)  by  means  of  a  Taylor  series.  This  expansion 
is  of  the  form 

/(z)  =/(Zo)  +/'(zo)  (z  -  Zo)  +  ^  (z  -  zo)^  +   .  •  .  , 

where,  as  we  have  seen,  /(zo)  =  0.  Not  all  of  the  derivatives  can  van- 
ish; for,  in  that  case  the  given  function /(z)  would  be  identically  zero  for 

*  See  Pierpont,  Theory  of  Functions  of  a  Reed  Variable,  Vol.  I,  p.  225. 


266  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

all  points  in  the  neighborhood  of  zo  and  hence  throughout  its  region 
of  existence.  The  first  non-vanishing  term  must  therefore  contain 
the  factor  (2  —  zo)  to  some  power,  say  the  A;"'  power.  We  have  then 
as  the  form  of  the  expansion 

/(«)  =  ak{z  -  Zo)''  +  ak+i{z  -  Zoy+'  +   •  •  •  . 

We  may  remove  the  factor  (2  —  2:0)*  from  each  term  of  the  series  and 
have 

f{z)  =  (z  -  Zo^iak  +  ak+i{z  -  Zo)  +   •  '  '  ]• 

Since  the  series  in  the  brackets  is  a  power  series  it  represents  some 
function  <j>{z)  which  is  holomorphic  in  the  neighborhood  of  zq.  More- 
over, <t>(zo)  =  Uk  is  different  from  zero.     Hence  we  have 

/(2)  =  (2  -  2o)V(2), 

where  4>{z)  satisfies  the  conditions  set  forth  in  the  theorem. 

The  point  2  =  20  is  said  to  be  a  zero  point  of  order  k  of  the  analytic 
function  f{z) ,  if  there  exists  a  positive  real  integral  value  of  k  such 
that  the  product 

^^^■^^'^ 

is  holomorphic  in  the  neighborhood  of  Zo  and  different  from  zero  for 
2  =  2o.  If  2  =  2o  is  a  regular  point  of  the  analytic  function  f{z)  and 
if  /(^)  =  0>  then  by  the  foregoing  theorem  2  =  zo  is  a  zero  point. 

By  multiplying  f(z)  by  the  factor  -, r^ ,  where  k  is  the  order  of 

\Z        Zo) 

the  zero  point,  the  vanishing  point  is  removed. 

That  a  theorem  does  not  exist  for  real  variables  analogous  to 
Theorem  II  for  analytic  functions  of  a  complex  variable  is  illustrated 
by  the  function 

f{x)  =  e   ^%        x^O. 

This  function  satisfies  the  conditions  of  Theorem  II,  stated  with 
reference  to  the  real  domain  in  the  deleted  neighborhood  of  the 
origin;  that  is,  it  has  all  derivatives  with  respect  to  x  in  this  deleted 
neighborhood,  and  moreover. 


L  fix)  =  0. 

x*0 


But  we  have  the  limit  * 


I     X* 
See  Stolz,  Differential^nd  Iniegrcdrechnung,  Part  I,  p.  81. 


Art.  51.]  SINGULAR  POINTS,   ZERO  POINTS  267 


for  all  values  of  k,  and  hence  the  zero  point  can  not  be  removed  by 

introducing  the  factor  -^,  no  matter  how  large  k  may  be  taken. 

Between  the  zero  points  and  the  poles  of  an  analytic  function, 
there  exists  the  following  relation. 

Theorem  III.     If  z  =  ZqIS  a  pole  of  order  k  of  the  analytic  function, 

f{z),  then  Ty-T  is  holomorphic  in  the  neighborhood  of  Zo  and  has  a  zero 

poird  of  order  k  aizo,  and  conversely. 

Since  z  =  2o  is  a  pole  of  order  k  oif(z),  we  have  from  the  definition 
of  a  pole 

(z  -  z,Yf{z)  =  0(2),  (3) 

where  4>{z)  is  holomorphic  in  the  neighborhood  of  Zq  and  <^(zo)  5^  0. 

Hence  in  the  neighborhood  of  Zq,  —j-ir  =  '^{z)  is  also  holomorphic 

4>{z) 

and  can  be  expanded  in  a  power  series,  the  first  term  of  which  is  a 
constant  different  from  zero.     From  (3)  we  have 

_L_  =  (,_^)*._^  =  (,_^)*.*(,).  (4) 

Since  both  {z  —  Zo)*  and  ^(2)  are  holomorphic  in  the  neighborhood 
of  2o  and  ^(zo)  5^  0,  the  last  member  of  (4)  is  holomorphic  in  the 
neighborhood  of  Zq  and  can  be  expanded  in  powers  of  {z  —  Zo)  begin- 
ning with  the  A;"'.     Hence  jtt  has  a  zero  point  of  the  order  k,  as 

f{z) 

stated  in  the  theorem. 

The  converse  of  the  foregoing  theorem  follows  similarly;    for,  if 

Tpr  is  holomorphic  with  a  zero  point  of  order  k  at  Zo,  we  may  write 

^  =  (2  -  ZoWiz),  (5) 

where  F{z)  is  holomorphic  and  different  from  zero  for  z  —  Zq.  Conse- 
quently, we  have 

(2  -  2o)V(2)   =  p^' 

where  -^rrx  is  also  holomorphic  and  different  from  zero  for  2  =  20. 

F{z) 

Hence  zo  is  a  pole  of  order  k  of  the  given  function /(2). 

In  the  foregoing  demonstration  we  have  made  use  of  the  fact 


^^ 


268  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

that  if  a  function  is  holomorphic  and  different  from  zero  in  the  neigh- 
borhood of  zq,  then  the  reciprocal  function  ?t-t-  is  also  holomorphic  and 

different  from  zero  in  the  neighborhood  of  zq.  If,  on  the  other  hand, 
the  point  Zo  is  not  a  regular  point  of  the  function /(s:),  then  this  point 
must  be  a  zero  point  or  an  essential  singular  point  of  the  reciprocal 
function  according  as  it  is  a  pole  or  an  essential  singular  point  of  f{z) . 

For  example,  z  =  0  is  an  essential  singular  point  of  e   *'  and  is  also,  an 

essential  singular  point  of  e  **. 

By  aid  of  Theorem  III  we  can  now  establish  the  following  theorem. 

Theorem  IV.  If  an  analytic  function  f{z)  is  holomorphic  and 
different  from  zero  in  the  deleted  neighborhood  of  z  =  Zq,  and  if 

L  f{z)  =  ^, 

then  the  point  z  =  Zois  a  pole  of  f{z). 

Since 

L  f{z)  =  00, 

we  have  at  once 

2=20  /(2) 

Asf{z)  is  holomorphic  in  the  deleted  neighborhood  of  Zo,  it  follows  that 

77-r-  is  holomorphic  and  different  from  zero  in  the  same  deleted  neigh- 

f{z) 

borhood.     Hence,  by  Theorem  II  there  exists  a  positive  integer  k  such 

that 

where  <f>(z)  is  holomorphic  in  the  neighborhood  of  Zo  and  (f>(zo)  ^  0. 
The  function  -jr-.  has  then  a  zero  point  at  Zo,  and  consequently  by 

Theorem  III,  /(z)  must  have  a  pole  at  the  same  point.  Hence  the 
theorem. 

As  in  the  case  of  Theorems  I,  II,  the  analogous  theorem  for  the 
realm  of  real  variables  does  not  exist,  as  the  following  illustration 
shows. 

Ex.  I.  Show  that  Theorem  IV  does  not  hold  for  the  following  function  (Fig.  97) 
of  a  real  variable,  namely: 

■  0 


Art.  51.] 


SINGULAR  POINTS,  ZERO  POINTS 


269 


The  conditions  of  Theorem  IV  are  satisfied  for  real  values  of  the  variable  in  the 
deleted  neighborhood  of  the  origin.  But  as  x  approaches  zero  the  product 
x^f{x)  becomes  infinite  *  for  all  values  of  k.  Hence,  the  infinity  of  the  fimction 
can  not  be  removed  by  introducing  the  factor  x*  no  matter  how  large  h  be  chosen. 


Y 


+  1 


O 


Fig.  97. 


-^X 


Theorem  V.     The  zero  points  of  an  analytic  function  are  isolated. 

Let  Zo  be  any  zero  point  of  the  analytic  function  f(z).    We  may 
then  write 

f{z)  =  (2  -  zo)*0(z),  (6) 

where  <^(z)  is  holomorphic  in  the  neighborhood  of  Zo  and  different 
from  zero  for  z  =  zq.  Since  <t>{z)  is  continuous,  we  can  then  draw  a 
circle  C  about  zo&s  a  center,  within  which  <l>(z)  does  not  vanish.  For 
any  value  oi  z  9^  Zo  within  C,  {z  —  ZqY  is  likewise  different  from  zero. 
Consequently,  within  C  there  is  no  point  other  than  Zq  at  which  f(z) 
vanishes.  The  zero  point  Zq  is  therefore  isolated.  But  Zo  was  any 
zero  point  of  f{z)  and  hence  all  such  points  are  isolated. 

*  See  Stok,  Differential-und  Integralrechnung,  p.  81. 


270  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

The  corresponding  theorem  for  the  poles  of  an  analytic  function 
may  be  stated  as  follows: 

Theorem  VI.  The  poles  of  an  analytic  function  are  isolaied  singu- 
lar points. 

If  an  analytic  function  f{z)  has  a  pole  at  any  point  Zo,  then  by 

Theorem  III  7^  is  holomorphic  in  the  neighborhood  of  20  and  has  the 

value  zero  at  Zo.  But  we  have  just  seen  (Theorem  V)  that  the  zero 
points  of  an  analytic  function  are  isolated.  Consequently,  the  poles 
must  also  be  isolated. 

If  the  poles  of  an  analytic  function  f(z)  have  a  limiting  point,  then 
the  behavior  of  f{z)  in  the  neighborhood  of  that  point  is  given  by  the 
following  theorem. 

Theorem  VII.  If  z  =  Zois  a  limiting  point  of  the  poles  of  an  ana- 
lytic function  f{z) ,  thenfiz)  has  an  essential  singularity  at  Zq. 

In  every  neighborhood  of  Zo  there  are  poles  of  the  given  function. 
The  point  z  =  Zo  can  not  then  be  a  regular  point  of  the  function,  and 
hence  must  be  either  a  pole  or  an  essential  singular  point.  It  can  not 
be  a  pole,  because  as  we  have  seen  (Theorem  VI)  every  pole  is  an 
isolated  singular  point.  It  must  then  be  an  essential  singular  point 
as  the  theorem  states. 

If  an  analytic  function  has  an  infinite  number  of  poles,  they  must 
have  at  least  one  Umiting  point  either  in  the  finite  region  or  at  infinity. 
We  now  see  that  at  this  limiting  point  the  function  has  an  essential 
singularity.  It  follows  then  that  an  analytic  function  having  no 
essential  singularities  can  have  but  a  finite  number  of  poles. 

We  have  seen  that  if /(z)  is  holomorphic  in  the  deleted  neighborhood 

of  Zo  and 

L/(0)  =  oo, 

then  2o  is  a  pole  of  fiz).  We  shall  now  show  that  conversely  an 
analytic  function  always  becomes  infinite  as  the  variable  approaches 
a  pole;  that  is,  we  shall  demonstrate  the  following  theorem. 

Theorem  VIII.  If  the  analytic  function  f(z)  has  a  pole  at  z  =  Zo, 
then  the  function  f{z)  always  becomes  infinite  o^  z  approaches  Zo  by  any 

path;  that  is, 

Lf(z)  =  oo. 


Abt.  51.]  SINGULAR  POINTS,  ZERO  POINTS  271 

Suppose  that  j{z)  has  a  pole  of  order  k  at  20,  then  by  the  definition 
of  a  pole,  we  have 

{z  -  z,YS{z)  =  <t>{z), 

where  <i>{z)  is  holomorphic  in  the  neighborhood  of  Zq  and  for  2  =  20  is 
different  from  zero.     Fof  values  oi  z  9^  zo,  we  have  then 

«^(2) 


m  = 


(2  -  20)* 


As  0(2)  is  finite  and  continuous  for  2  =  20,  then  as  2  approaches  26 

th(z) 

by  any  path  whatsoever  y rr  increases  in  absolute  value  without 

{Z  —  Zq)  I 

limit.     Consequently,  we  may  write 

Lf(z)==<x>, 

as  the  theorem  requires. 

Not  only  may  an  essential  singular  point  of  an  analytic  function 
appear  as  a  limiting  point  of  poles,  but  it  may  also  be  the  limiting 
point  of  other  essential  singular  points  or  it  may  appear  as  an  iso- 
lated singular  point  of  the  function.  For  isolated  essential  singular 
points,  that  is  essential  singular  points  that  are  not  the  limiting  points 
of  other  essential  singular  points,  we  have  the  following  theorem.* 

Theorem  IX.  If  Zq  is  an  isolated  essential  singular  point  of  f{z), 
and  /3  is  any  arbitrary  number,  real  or  complex,  then  2  may  he  made  to 
approach  zq  in  such  a  manner  that  the  corresponding  values  of  f{z)  have 
the  limiting  value  jS. 

By  hypothesis  the  point  Zq  can  not  be  the  limiting  point  of 
other  essential  singular  points  of  the  function,  although  it  may  be 
the  limiting  point  of  poles  of  the  function.  Moreover,  there  can  not 
exist  a  neighborhood  of  20,  however  small,  such  that  at  every  point 
of  it  we  have  f{z)  =  /S;  for,  in  this  case  f{z)  would  have  by  Theorem  I 
at  most  a  removable  discontinuity  at  20,  and  hence  this  point  could 
not  be  an  essential  singular  point.     Consider  the  function 

for  values  of  2  in  the  neighborhood  of  Zq. 

*  This  theorem,  commonly  attributed  to  Weierstrass,  was  doubtless  first 
demonstrated  by  the  Italian  mathematician,  Casorati. 

See  Rend.  1st.  Lomb.,  (2)  I,  1868;  also  Vivanti-Gutzmer,  Theorie  der  ein- 
deiUigen  analytischen  Funktionen,  p.  130. 


272  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

Either  there  exists  a  finite  number  M  such  that  for  all  values  of  z 
in  every  neighborhood  of  zq  we  have 

l^(^)l<M, 

or  there  exists  no  such  number  M.  We  shall  show  that  the  first 
of  these  alternatives  is  impossible  under  the  conditions  of  the  theorem. 
In  fact,  if  a  finite  number  M  can  be  found,  then  by  Theorem  I,  Zo 
is  a  regular  point  of  F{z).  If  in  addition  F(z)  is  equal  to  zero  for 
z  =  Zo,  then  Zq  is  by  Theorem  III  a  pole  of /(z)  —  )3  and  hence  oi  f{z). 
If,  on  the  other  hand,  F{z)  is  not  equal  to  zero,  then  Zo  is  a  regular 
point  of  f(z)  —  /3  and  hence  oi  f(z).  But  either  of  these  conclusions 
is  a  contradiction  to  the  given  hypothesis,  for  Zo  is  by  the  conditions 
of  the  theorem  an  essential  singular  point  of  f{z).  It  follows  that 
there  can  exist  no  finite  value  of  M  such  that  for  all  values  of  z  in 
every  neighborhood  of  Zo  we  have  |  F{z)  \  <  M. 

In  every  neighborhood  of  Zo  however  small  there  are  then  points  at 
which  \F{z)  \  >  M,  however  large  M  may  be  taken;  that  is,  in  every 

neighborhood  of  Zo  there  are  values  of  z  for  which  |  F{z)  \  >- ,  where  e 

is  arbitrarily  small.     For  all  such  values  we  have 

1/(2)  -PI  <e. 

We  may  then  select  a  set  of  points 

Zl,  Z2,  .   .  .  ,  z„,   .   .   .  , 

having  Zo  as  a  limiting  point,  such  that 

L    f(Zn)    =  p. 

The  foregoing  theorem  must  not  be  understood  to  mean  that  in 
the  neighborhood  of  an  essential  singular  point  /(z)  actually  takes 
every  value.  Picard  has  shown,*  however,  that  in  the  neighborhood 
of  an  essential  singular  point  a  single-valued  analytic  function  takes 
all  complex  values  with  the  exception  of  at  most  two  and  indeed  an 
infinite  number  of  times. 

Thus  far  we  have  considered  only  those  singularities  of  a  single- 
valued  function  that  occur  at  finite  points  of  the  complex  plane. 
To  determine  the  nature  of  the  function  in  the  neighborhood  of  the 

*  See  M&moire  sur  Us  fonctions  entihres,  Ann.  de  I'Ecole  Normale,  1880; 
also  Traiti  d'analyae,  Vol.  II,  p.  121. 


Art.  51.]  SINGULAR  POINTS,   ZERO   POINTS  273 

infinite  point,  we  subject  the  variable  z  to  the  reciprocal  transforma- 
tion 

1 

2  =  - 
z 

and  examine  the  transformed  function  0(2')  for  values  of  z'  in  the 
neighborhood  of  the  point  z'  =  0.  The  given  function  /(z)  is  said 
to  have  a  pole  or  an  essential  singularity  at  infinity  according  as 
2'  =  0  is  a  pole  or  an  essential  singular  point  of  <l>(z').  The  function 
(2)  is  said  to  have  a  regular  point  at  infinity  if  2'  =  0  is  a  regular 
point  of  the  function  0(2'). 

In  case  the  point  2  =  go  is  a  regular  point  of  the  given  function 
f{z),  then  the  transformed  function  <^(2')  is  holomorphic  in  the  neigh- 
borhood of  2'  =  0.  We  can  then  expand  ^(2')  in  a  Maclaurin  series 
and  have 

<^(z')  =00  +  aiz'  +  •  •  •  +  a„2'"  +  •  •  •  . 

Consequently,  the  expansion  of  f{z)  in  the  neighborhood  of  2  =  00 , 
when  this  point  is  a  regular  point  of  the  function,  is  of  the  form 

It  has  been  shown  that  if  f{z)  is  holomorphic  in  a  given  region  S, 

then  the  integral    /  f{z)  dz  taken  around  any  closed  curve  C  lying 

wholly  within  S  and  inclosing  only  points  of  S  must  vanish.  It  is 
of  interest  in  this  connection  to  point  out  that  this  conclusion  does 
not  hold  when  the  given  region  includes  the  point  at  infinity.  For 
this  case,  we  have  the  following  theorem. 

Theorem  X.  //  C  is  an  ordinary  curve  inclosing  the  point  at  infin- 
ity and  lying  within  a  given  region  S  which  likewise  contains  the  point 

al  infinity,  then  the  integral  j  f{z)  dz  vanishes  if  z^  f{z)  is  holomorphic 

in  S. 

Putting  2  =  — ,  we  have 

ff{z)  dz=  -  f/-^  <!>(_/)  dz',  (Art.  22) 

Jc  Jy 

where  7  is  the  curve  about  the  origin  into  which  the  curve  C  is  mapped 
by  the  transformation  2  =  — .    The  given  integral  vanishes  whenever 


274  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

the  integral  —    /  z'~^  <f>{z')  dz'  vanishes,  that  is,  if  z'~^  <l>{z')  is  holo- 

morphic  in  a  region  S'  about  the  origin  within  which  the  curve  7  hes. 
However,  if  z'~^  4>{z')  is  holomorphic  in  S',  then  2^(2)  must  be  holo- 
morphie  in  the  corresponding  region  S  about  the  point  infinity  and 
conversely.    Hence  the  theorem. 

Theorem  XI.  The  circle  of  convergence  of  a  power  series  passes 
through  at  least  one  singular  point  of  the  analytic  function  determined  by 
the  series. 

In  the  discussion  of  Taylor's  series  (Art.  48)  it  was  pointed  out  that 
the  power  series  in  (z  —  Zo)  resulting  from  the  expansion  of  a  given 
function  which  is  holomorphic  in  a  region  *S  converges  and  represents 
that  function  for  all  values  of  z  within  any  circle  that  can  be  drawn 
about  Zo  as  long  as  it  lies  within  S  and  incloses  only  points  of  S. 
That  is,  the  size  of  the  circle  C  (Fig.  87)  within  which  the  series  is 
known  to  converge  is  limited  only  by  the  region  S  in  which  the  given 
function  is  holomorphic.  As  we  now  know,  that  region  S  is  restricted 
only  by  the  presence  of  singular  points  of  the  analytic  function  f{z) 
of  which  the  given  power  series  defines  an  element.  Consequently, 
if  the  circle  C  is  the  circle  of  convergence  of  the  Taylor  series,  then  it 
must  pass  through  at  least  one  singular  point  of  f{z) ;  that  is,  within 
every  larger  concentric  circle  there  must  be  at  least  one  such  point; 
otherwise,  a  larger  circle  than  C  might  be  selected  in  determining 
a  region  within  which  the  Taylor  series  converges  and  the  series  would 
then  converge  for  points  outside  the  circle  C.  This  circle  would  not 
then  be  the  circle  of  convergence  as  assumed. 

If  a  function  is  holomorphic  in  a  given  region,  it  can  be  expanded 
in  a  Taylor's  series  for  values  of  z  in  the  neighborhood  of  any  point 
of  that  region.  The  expression  obtained  for  the  coefficients  of  such 
an  expansion  enables  us  to  establish  the  following  theorem,  due  to 
Liiouville. 

Theorem  XII.  A  single-valued  analytic  function  which  has  no 
singularity  either  in  the  finite  portion  of  the  plane  or  at  infinity  redvxxs 
to  a  constant. 

If  a  function  f{z)  has  no  singularity  either  in  the  finite  region  or  at 
infinity,  it  follows  that  it  is  everywhere  less  in  absolute  value  than 
some  definite  number  M;  for,  otherwise,  there  would  exist  a  point  Zo, 
finite  or  infinite,  in  every  neighborhood  of  which  f{z)  would  exceed 


Art.  52.]  LAURENT'S  EXPANSION  275 

in  absolute  value  all  finite  bounds,  that  is,  would  become  infinite. 
The  function  admits  of  a  Maclaurin  expansion  about  the  origin, 
namely 

oo  +  ai2:  +  a22^+   •  •  •   +  an^"  +   •  •  •  (6) 

which  converges  and  represents  the  function  for  all  finite  values  of  z. 
The  coefficient  a„  is 


nr    (°>      2inJc   z"+^  ' 


where  C  is  a  circle  of  radius  p  about  the  origin  as  a  center,  the  value 
of  p  being  taken  as  large  as  we  please.     Since 

\A^)\<M, 
we  may  write  ,     , 

I       ,1     CMldz]      1      M     ^  M 

'  ""  '  <  2^  Jc7^=  2^'p^-2"^  =  ^' 

Inasmuch  as  p  can  be  taken  as  large  as  we  choose,  it  follows  that 

\an\  <  e,  n>  0, 

where  e  is  an  arbitrarily  small   positive  number.     Consequently, 
since  «„  is  a  constant,  we  must  have 

an  =  0,         n  =  1,2,3,  ...  . 
It  follows  from  equation  (6)  that 

f(z)  =  «o 

for  all  values  of  z  in  the  finite  portion  of  the  plane.     Since  the  point 
2  =  GO  is  a  regular  point  of  f{z),  we  have 

f{z)  =   L  J{z)  =  ao. 

It  follows  from  the  foregoing  theorem  that  every  single-valued 
analytic  function  which  is  not  a  constant  must  have  at  least  one  singu- 
lar point  either  in  the  finite  portion  of  the  plane  or  at  infinity, 

52.  Laurent's  expansion.  We  have  seen  that,  in  the  neighbor- 
hood of  a  regular  point  of  an  analytic  function,  it  can  be  represented 
by  a  power  series,  but  this  method  of  representation  does  not  hold 
in  the  neighborhood  of  a  singular  point  of  the  analytic  function.  We 
shall  now  show  that  in  the  neighborhood  of  an  isolated  singular  point 
2o  we  can  expand  an  analytic  function  in  a  series  having  also  negative 
powers  of  {z  —  Zo).  Such  a  series  is  not  properly  a  power  series,  since 
a  power  series  was  defined  as  a  series  involving  only  positive  integral 
powers.  We  shall,  however,  often  refer  to  the  series  involving  nega- 
tive powers  as  a  power  series  with  negative  exponents  or  a  power 


276 


SINGLE-VALUED  FUNCTIONS 


[Chap.  VII. 


senes  m 


1 


z  —  Zo 


When  the  term  power  series  is  used  without  a 


Fig.  98. 


qualifying  phrase,  we  shall  as  heretofore  understand  it  to  mean  a 
series  involving  only  the  positive  powers  of  the  variable. 

In  the  derivation  of  Taylor's  expansion  (Art.  48),  it  was  found  to 
be  valid  within  a  region  bounded  by  a  single  circle,  provided  there 

^  are  no  singular  points  of  the  given 
analytic  function  within  the  circle. 
Suppose  we  now  consider  a  region 
S  bounded  by  two  concentric  circles 
Ci,  Ci  (Fig.  98)  such  that  within  S, 
f(z)  has  no  singular  points  and  con- 
verges uniformly  to  finite  values 
along  each  circle.  There  are  no 
restrictions  as  to  singular  points 
exterior  to  Ci  or  interior  to  C2. 
Denote  the  common  center  of  Ci, 
Ci  by  Zq.  To  apply  this  method 
later  to  the  expansion  of  a  function 
in  the  neighborhood  of  a  singular  point,  it  is  convenient  to  take  the 
radius  of  d  arbitrarily  small. 

As  in  the  consideration  of  Taylor's  series,  we  shall  base  our  dis- 
cussron  upon  the  fact  that  we  can  express  the  given  analytic  function 
f{z)  by  means  of  the  Cauchy  integral  formula.  Since  the  given  region 
S  is  bounded  by  two  curves  the  integral  must  be  taken  over  the  entire 
boundary  and  hence  along  the  two  curves  in  the  directions  indicated 
in  the  figure.  Taking,  however,  the  integral  along  C2  in  a  negative 
direction  with  respect  to  the  region  S,  that  is  in  a  counter-clockwise 
direction,  we  have  for  any  value  of  2  in  S 

-'^^      2TdJc,t-z      2TdJc,t-z'  ^^ 

where  t  is  taken  along  each  of  the  curves  Ci,  C2  in  a  counter-clockwise 
direction.  Since  z  is  any  point  in  S,  then  for  the  first  integral  we 
have  \z  —  Zo\<\t  —  zq\.  By  Theorem  III,  Art.  20,  this  integral 
by  itself  defines  a  function  <^(2)  which  is  holomorphic  for  all  values 
of  2  within  Ci  and  hence  can  be  expanded  in  a  power  series  in 
(2  —  2o)  by  means  of  Taylor's  expansion.  Such  an  expansion  is  of 
the  form 


<i>{z)  =  oo  -f-  ai(2  —  2o)  +  02(2  —  2o)2-|- 


-I-  an{z  -  20)"+ 


(2) 


Art.  52.]  LAURENT'S  EXPANSION  277 

where  we  have 

|.-e,|<|(-.,|,    a.  =  2^^^^^M|_,       „  =  o,       1,2,-... 

The  second  integral  also  defines  a  function  r/'(0)  which  is  holomorphic 
for  values  exterior  to  d,  that  is  for  \  z  —  Zo  \  >  \  t  —  zo  \,  the  values  of  t 
being  limited  to  values  on  C2.  To  find  a  form  of  expansion  for  \l/(z),  we 
proceed  in  a  manner  similar  to  that  used  in  the  discussion  of  Taylor's 

series.    We  shall  consider  the  function  ■-, ,  which  occurs  in  the 

t  —  z 

given  integrand.     We  may  write 

1  1       A-^o\  -1     I  1 

t  —  Z         Z  —  ZQ\t  —  Z  )        Z  —  Zq\  t  —  Zq 

I  Z  —  Zq 

1  t-Zo  (t-  ZoY  {t  -  0o)"-' 


2  —  20         (2  —  2o)2         (2  —  ZoY  (2  —  2o)" 


(3) 


This  series,  considered  as  a  series  in  t,  converges  uniformly  (Art.  45, 
Theorem  I)  for*  any  constant  value  of  2  such  that  \z  —  Zo\>\t  —  Zo\, 
that  is  for  any  value  of  2  exterior  to  C2.  The  property  of  uniform 
convergence  is  not  destroyed  by  multiplying  each  term  of  (3)  by 
fit).    We  have  then 

m^   m   (t-zo)m  {t-z,ym    (^-2o)"-y(o 

t—z        z—Zq      {z—ZqY         {z—ZoY  (2—20)" 

Since  this  series  converges  uniformly,  we  may  integrate  it  term  by 
term,  thus  obtaining 


27rt 


4^,X_«-^)"-W)<«+---* 


The  integrals  in  the  second  member  of  this  equation  determine  the 
coefficients  of  the  desired  expansion  of  the  second  integral  in  (1). 
We  may  therefore  write 
^(2)  =  a-i(2-2o)-i  +  a-2(2-2o)-2+  •  •  •  +a_„(2-2o)-"+   •  •  •   ,      (4) 

where 

|2-2ol>  U-20I,     a-n=^.  f  (t-Zi^r-'mdt,    n=l,  2,  •  •  . . 


278  SINGLE-VALUED   FUNCTIONS  [Chap.  VII. 

Since  the  series  (2)  converges  for  all  values  of  z  within  C\  and  the 
series  (4)  for  all  values  of  z  exterior  to  Ca,  it  follows  that  both  con- 
verge for  values  of  z  within  S>  bounded  by  these  two  circles.  Conse- 
quently for  values  of  z  within  *S,  the  given  function /(z)  may  be  written 
as  the  sum  of  two  functions  <^(z),  ^(z),  the  first  of  which  can  be  ex- 
panded in  a  series  involving  the  positive  integral  powers  of  (z  —  Zo), 
and  the  second  of  which  can  be  expanded  in  a  series  involving  the 
negative  integral  powers  of  (z  —  Zo);   that  is,  we  have 

00  00 

/(Z)  =  «/>(z)+  ^(Z)  =  X  «n(2  -  Zo)"  +  X  «-»(^  -  ^o)""- 
n=0  n=l 

We  may  replace  the  two  circles  Ci,  Ci  as  paths  of  integration  by  a 
single  path  of  integration.  This  path  of  integration  may  be  any 
ordinary  closed  curve  C  lying  within  ;S,  and  inclosing  Ci  since  each 
of  the  circles  Ci,  C2  may  be  deformed  into  C  without  passing  over  a 
singular  point  of  the  integrand.  The  coefficients  of  the  two  series 
(2)  and  (4)  may  then  be  expressed  in  terms  of  the  integrals  taken  over 
the  curve  C.    We  have  then  the  following  theorem. 

Theorem  I.  ///(z)  is  holomorphic  in  the  annular  region  S  bounded 
by  two  concentric  circles  ohout  a  given  point  Zo,  then  vrithin  this  region 
f(z)  can  be  represented  by  a  series  of  the  form 

00 
X«»(2-Zo)",  (5) 

—  00 

where 

"'•  =  9l7  f(t-Zo)-"-'f(t)dt, 

and  C  is  any  ordinary  curve  lying  whoUy  within  S  and  inclosing  the 
inner  circle. 

The  series  (5)  is  known  as  Laurent's  series.  While  there  may  be 
an  infinite  number  of  terms  of  the  series  corresponding  to  negative 
values  of  n,  on  the  other  hand  only  a  finite  number  of  such  terms  may 
appear  in  the  expansion,  the  number  depending  as  we  shall  see  upon 
the  character  of  the  function  /(z)  at  the  point  Zq.  By  aid  of  the  fore- 
going theorem  we  can  now  represent  a  single-valued  analytic  func- 
tion in  the  deleted  neighborhood  of  an  isolated  singular  point  by 
means  of  a  series  involving  the  positive  and  negative  powers  of  the 
variable;  for,  if  Zo  is  such  a  singular  point,  then  by  making  the  radius 
of  Cz  sufficiently  small  but  different  from  zero  we  can  include  in  the 


Art.  52.1  LAURENT'S  EXPANSION  279 

region  S  any  point  in  the  deleted  neighborhood  of  Zq.     Hence,  while 

00  00 

^  aniz  —  2o)"  converges  for  all  values  of  z  within  Ci,  ^a-niz  —  Zo)"" 

n=0  n=l 

converges  for  all  values  of  z  within  Ci  except  z  =  Zq. 

As  has  already  been  pointed  out,  the  nature  of  a  singular  point  of 
an  analytic  function  is  fully  determined  by  the  behavior  of  the  func- 
tion in  the  deleted  neighborhood  of  that  point.  The  Laurent  expan- 
sion of  the  function  also  determines  the  character  of  the  singularity. 
For  example,  ii  z  =  Zo'isa,  pole  of  order  k  of  the  analytic  function /(«), 
then  we  are  able  to  remove  the  singularity  by  multiplying  f{z)  by 
the  factor  (z  —  Zo)*.  Hence  there  are  k  terms  in  the  Laurent  expan- 
sion having  negative  exponents;  that  is,  the  expansion  is  of  the  form 

-,  ,  a-k        .        oc-k+i        I  ,        a-i        11/  \ 

+  a^iz  -  Zo)2  +  .  •  .  +an(2  -  Zo)"  +  •  •  .  .  (6) 
That  part  of  the  expansion  which  indicates  the  character  of  the  singu- 
larity, namely 

—k 

2   Ctr{Z  -  ZoY, 
r=-l 

is  called  the  principal  part  of  the  expansion.     In  case  of  a  pole  of 
order  k,  it  consists  of  k  terms. 

The  Laurent  expansion  of  a  given  analytic  function  in  the  neigh- 
borhood of  an  isolated  singular  point  may  be  accomplished  by  direct 
application  of  Theorem  I,  but  if  the  singular  point  Zo  is  a  pole  of 
order  k,  we  may  write 

where 

<f)(z)  =  a-k  +  a-k+i(z  —  Zo)  +   •  •  •  ,     a-k  9^  0, 
whence 

/(^)  =  (^z:^.  +  (^ir^pi+ •••  + (^^r^+«o+«^(^-^'>)  +  •  •  •  • 

Ex.  1.    Expand  the  function 

in  a  series  for  values  of  z  in  the  neighborhood  of  the  origin. 
This  function  can  be  written  in  the  form 

/w  -  -^' 

where  ^(z)  =  -. =  -4  2  + 


1-2  ' 1 -z 

=  1-32+ 22  +23+    .    .    .    . 


\ 


280  SINGLE-VALUED  FUNCTIONS  [Chap.  VIL 

Hence,  we  have 

^=^-1  +  1  +  ^+^+^^'+^+  ■■•  ■ 

13      1 

The  terms  -z  — ;  +  -  are  the  principal  part  of  the  expansion  of  /(z)  in  the  neigh- 
z*     z*      z 

borhood  of  the  origin. 

If  f(z)  has  an  isolated  essential  singular  point  at  Zo  which  is  not 
the  limiting  point  of  poles  of  f(z),  then  there  is  no  finite  value  of  k 
such  that  (z  —  zoYfiz)  is  holomorphic  in  the  neighborhood  of  that 
point,  and  hence  the  Laurent  expansion  has  an  infinite  number  of 
terms  involving  negative  powers  of  {z  —  Zo).  The  expansion  is  then 
of  the  form 


{z  —  Zo)*       {z  —  2o)*~^  z  —  Zo 

+  00  +  ai{z  -  2o)  +   .  .  .   +  an(z  -  2o)"  +   •  •  •   I       (7) 

that  is,  the  principal  part  of  the  expansion  consists  of  an  infinite  num- 
ber of  terms. 

In  case  zo  is  a  regular  point  of  the  function,  the  Laurent  expansion 
has  no  terms  with  negative  exponents  and  hence  becomes  identical 
with  the  Taylor  expansion. 

If  the  point  z  =  cx)  is  a  pole  of  order  A;  of  a  given  function  f{z),  then 
the  function  <l>(z')  obtained  by  transforming  f{z)  by  the  relation 

z  —  —  must  have  a  pole  of  order  k  at  the  origin.     Its  expansion  is 
z 

therefore  of  the  form 

•^(^O  =$r  +  5S^+   •  •  •   +y-'+«o+ai/+   •  •  •  +an2'"+   •  ••  . 

In  the  neighborhood  of  z  =  oo  the  expansion  of  /(z)  is  therefore  of 
the  form 

/(z)=a_fc2*  +  a_i+iz*-'+   .  .  .  +a_i2+ao+^+   .  .  .   +^+   .  .  . 

Z  2" 

the  first  k  terms  constituting  the  principal  part. 

As  with  Taylor's  expansion,  the  question  arises  as  to  whether  an 
analytic  function  is  uniquely  represented  by  means  of  a  Laurent 
series.  In  this  connection  we  may  well  consider  the  following 
theorem. 

Theorem  II,  7/  in  an  annular  region  S  a  given  function  f(z)  per- 
mits of  an  expansion  of  the  form 


fiz)  =  2  an (2  -  2o)", 


Abt.  52.]  LAURENT'S  EXPANSION  281 

then  the  coefficients  of  this  expansion  are  given  by  the  relation 

«n  =  2^j^(<-2o)-"-V(0d<; 
thai  is,  there  is  but  one  such  expansion  possible. 

As  in  the  discussion  of  Theorem  I,  let  the  region  aS  be  bounded  by 
the  circles  C],  C2,  having  the  point  20  as  a  center.  We  assume  the 
existence  of  an  expansion  of  f{z)  in  a  series  as  stated  in  the  theorem. 
Denote  by  C  any  circle  concentric  with  Ci,  C2  and  lying  between  the 
two,  and  let  the  value  of  the  variable  along  C  be  denoted  by  t.  The 
given  series  converges  along  C  and  expressed  in  powers  of  (t  —  Zq)  is 

+  ao  +  ai  («  -  Zo)  +   •  •  •   +(Xn{t-  2o)"  +   •  •  •  .      (8) 

This  series  converges  uniformly  (Theorem  I,  Art.  45)  and  hence  may 
be  integrated  term  by  term  along  C.     Before  doing  so,  however,  let 

us  multiply  the  terms  of  the  series  by  the  factor  j- r^-    Remem- 
bering that 


£ 


(t  -  ZoYdt  =  0,        n9^  -1,     (Exs.  1,  2,, Art.  18) 

=  27rt,    n  =  -1,  (9) 

it  follows  that  the  integrals  of  all  of  the  terms  of  the  series  vanish 
except  one,  namely  the  term  involving We  have  then  as  the 

t  —  Zq 

result  of  the  integration 

mdt 

=  Zman, 


Jc^ 


whence,  we  have 

«n  =  ^.     I  {t-Zo)—'mdt,  (10) 

which  establishes  the  theorem. 

It  does  not  follow  from  what  has  been  said  that  J{z)  may  not  have 
different  Laurent  expansions  in  different  circular  regions.  For  ex- 
ample, suppose  we  have  two  regions  (Fig.  99)  one  bounded  by  the 
circles  C1C2  and  the  other  by  C2C3,  where  C2  has  upon  it  a  singular 
point  of  the  given  function.  In  each  of  these  regions  there  is  an 
expansion  in  a  Laurent  series,  but  the  two  expansions  in  such  a  case 
are  not  identical.  This  condition  is  illustrated  by  the  following 
example. 


282  SINGLE-VALUED  FUNCTIONS 

Ex.  2.    Given  the  function 


nz)  = 


1 


1 


2*-32+2        2-2        2-1 


[Chap.  VII. 


(11) 


Fig.  99. 

This  function  has  two  poles,  namely,  z  =  1,  2  =  2.  Within  the  circle  about  the 
origin  passing  through  the  point  2  =  1,  the  function  can  be  represented  by  a 
Maclaurin  series.    The  resulting  series  is 


2  +  i"  +  8-^  +16"  +  •••  + 


2"+i 


2"  + 


which  converges  and  represents  the  given  function  for  |  z  |  <  1. 

If  we  take  1  <  1  2  |  <  2,  we  must  use  Laurent's  expansion.     The  coeflBcients 
of  the  series  are  given  by 

where  C  is  any  circle  about  the  origin  lying  between  Ci  and  C2. 
Putting  for/(0  its  value  from  (11),  the  integrand  in  (12)  becomes 

1 1 

Upon  decomposing  each  of  these  fractions  into  partial  fractions,  we  have  for 
n  S  0 


I   I 

f     1. 
[t-2 

1 
t-1 

2"  I 

tn  +  ij 


Replacing  the  integrand  in  (12)  by  these  partial  fractions,  since 

f  C'dt  =0,        Ut^  -\, 

=  27rt,     n  =  —  1, 
we  have 

^      1       r    dt 1 ]_  r     dt 

""      2^+hriJct-2      2"+i      2inJct-l'^ 


(13) 


Art.  52.]  LAURENT'S  EXPANSION  283 

But ^  is  holomorphic  within  C  and  hence  the  intecral    f vanishes. 

t  —  2  ^      Jc  t  —  2 

To  evaluate  the  second  integral  in  (13)  we  deform  the  path  C  of  integration  into 

a  small  circle  C  about  the  point  z  =  1  and  put  ^-^.^  ,_^ 

t-1  =  pei9,  ( 

where  p  is  a  constant  and  d  varies  from  0  to  2  tt.     We  have  then 

Jet  —  1      Jet  —  1         Jo 
Consequently  from  (13),  we  have 

-  _  J_  _i  a-i  1 

The  terms  of  the  required  Laurent  expansion  corresponding  to  values  of  n  =  0 
are  then 

-I'V-y -2^1^" •  (14) 

For  negative  values  of  n,  say  w  =  —k,  we  have 

2id  \_Jc  t-2        Jc  t-l  A 

But  the  integrand  in  the  first  integral  is  holomorphic  within  C  and  hence  the 
integral  vanishes.     We  have  by  deforming  C  into  C  and  putting  as  before 
t-\  =  pe^, 

"""*  2iriJo  pe»9  r 

J       (l+peie)*-ide        [  V 


=   -;r^.    I         d«=   -1.  -1 

2iri  Jo 


Hence,  each  term  in  that  portion  of  the  Laurent  expansion  having  negative  ex- 
ponents has  the  coefficient  —  1.    The  complete  expansion  is  therefore 

1  1      1       1_1     _1^_  L_   n_  nn\ 

...___...       ^^      ^      ^      ^2      gg.       ...       ^^^2       ....    ^a; 

It  is  evident  that  the  given  function  has  no  finite  singular  point  exterior  to 
the  circle  about  the  origin  and  passing  through  the  point  2  =  2.  The  expansion 
of  the  function  about  the  point  z  =  oo  will  then  hold  for  this  entire  region.  By 
putting 

'  =  ^' 

the  entire  region  exterior  to  the  circle  C2  through  2  =  2  inverts  into  the  region 
about  the  origin  and  Ijdng  within  the  circle  C"  whose  radixis  is  3.  The  trans- 
formed function  is 

z'  z' 


284  SINGLE-VALXJED  FUNCTIONS  [Chap.  VII. 

Within  the  circle  C"  the  function  4t{z')  is  holomorphic  and  may  be  expanded  in  a 
Maclaurin  series,  giving 

4>{z')  =  z'2  +  3  2'»  +  7  2'*  +  •  •  •  . 

^Replacing  z'  by-,  we  have  as  the  expansion  of  the  given  function  for  values  of 
s  exterior  to  the  circle  Ci, 

i+3+_7  +2n-^-l 

The  same  result  would  have  been  obtained  had  we  expanded  the  function  by 
computing  the  coefficients  by  aid  of  the  formula  given  in  Theorem  II,  where 
the  path  of  integration  is  any  circle  Cz  about  the  origin  and  lying  exterior  to  the 
concentric  circle  through  2  =  2. 


53.  Residues.     We  have  seen  that  the  integral    /  f{z)  dz 


van- 


ishes when  taken  around  the  boundary  C  of  a  region  <S,  provided  f{z) 
is  holomorphic  in  the  open  region  S  and  at  least  converges  uniformly 
to  its  values  along  C.  Let  us  now  consider  the  effect  upon  this  inte- 
gral when  S  contains  one  isolated  singular  point  of  J{z).  Before 
doing  so,  we  introduce  the  following  definition. 

If  the  points  of  ^S,  with  the  exception  of  at  most  the  point  Zq,  are 
regular  points  of  j{z)  and  C  is  any  closed  curve  about  Zo  and  lying 
wholly  within  S  and  aside  from  z^  containing  only  points  of  S,  then 

the  integral 

1     r 

dz 


^■//« 


taken  in  the  positive  direction  is  called  the  residue  of  f{z)  at  Zq. 

Suppose  the  point  Zo  is  a  pole  of  the  given  function.  We  have  then 
the  following  theorem. 

Theorem  I.  Iff(z)  is  holomorphic  in  a  given  finite  region  S  except  at 
Zo,  where  it  has  a  pole,  then  the  residue  off(z)  at  Zo  is  equal  to  the  coefficient 
of  {z  —  2o)~^  in  the  expansion  of  f{z)  in  powers  of  {z  —  Zq). 

Let  the  pole  at  Zo  be  of  order  k.  Then  the  Laurent  expansion  of 
the  function  in  powers  of  (z  —  Zo)  is  of  the  form 

'     -(z-zo)*  +  (z-zo)*-^+  •••  -^^zr^  +  <i>(^)> 

where  <f>(z)  is  holomorphic  in  the  neighborhood  of  Zo,  say  within  and 


Art.  53.]  RESIDUES  285 

upon  a  circle  C  having  Zo  as  a  center.     Taking  C  as  the  path  of  inte- 
gration, we  have 

ff(z)  dz  =  a-k  f  (z-  zoj-^dz  +  •  •  . 
Jc  Jc 

+  a-ij  (z  -  zo)-'  dz+  j  <t>(z)  dz.  (1) 

The  integral  />  (f)(z)  dz  vanishes,  since  ^(z)  is  holomorphic  in  the  closed 

region  bounded  by  C.     To  evaluate  the  remaining  integrals  we  make 
use  of  the  relations 

/  (z-  zo^dz  =  0,  n9^  -1, 

Jc 

=  27rt,         n  =  -1.  (2) 

Consequently,  we  have  from  (1) 

Jf{z)  dz  =  27ria_], 
c 

whence  a-i  =  ^r— .  /  f(z)  dz,  (3) 

which  establishes  the  theorem,  since  by  definition  the  second  member 
of  this  equation  is  the  residue. 

The  value  of  the  residue  of  an  analytic  function  at  a  pole  is  zero  if 
the  coefficient  a_i  is  zero  in  the  expansion  of  the  function.  For 
example,  the  function 

has  a  pole  of  order  three  at  the  origin,  yet  the  residue, 

1      Cdz 

2  Trt  Jc  2^ 

is  zero. 

The  foregoing  theorem  gives  the  residue  when  there  is  a  single 
isolated  pole  in  the  given  region  S.  If  there  are  a  finite  number  of 
poles  in  S,  we  have  the  following  theorem. 

Theorem  II.  Given  a  function  f{z)  which  is  holomorphic  in  a 
region  S  with  the  exception  of  a  finite  number  of  poles,  and  let  C  be  any 
ordinary  curve  lying  wholly  within  S  and  inclosing  all  of  the  given 

poles.     Then   I  f{z)  dz  taken  in  a  positive  direction  is  equal  to  2  tti  tim£s 
J  c 

the  sum  of  the  residues  of  f{z)  at  these  poles. 


■i 


286  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

Suppose  the  poles  of  /(z)  to  be  Zi,  22,  •  •  .  ,  2*,  .  .  .  ,  z„.  About 
each  of  these  points  as  a  center  draw  an  arbitrarily  small  circle  lying 
wholly  within  the  region  bounded  by  C.  Denote  these  circles  by  Ci, 
C2,  .  .  .  ,  Cjfc,  .  .  .  ,  Cn.     Then  by  Theorem  VI,  Art.  19,  we  have 

ff{z)dz  =  X   fs{z)dz.  (4) 

But  as  we  have  seen    /  /(z)  dz  is  equal  to  2  Tri  times  the  residue  of 

/(z)  at  Zk  and  is  given  by  the  coefficient  of  (z  —  z*)"^  in  the  Laurent 
expansion  of  /(z)  in  powers  of  (z  —  Zk).  The  relation  given  in  (4) 
therefore  establishes  the  theorem. 

A  closed  curve  C  may  be  regarded  as  the  boundary  of  either  of  two 
regions,  one  finite  and  the  other  inclosing  the  point  at  infinity.  It  is 
readily  seen  that  the  relation  (4)  holds  when  C  is  regarded  as  bounding 
the  outer  region^as  well  as  in  the  case  just  considered.     Theorem  II  is 


still  valid  therwj  (^herv f{z)  dz  is  taken  over  a  curve  inclosing  the  point 

2  =  00.  However,  when  the  point  z  =  00  is  a  pole  the  residue  at 
that  point  is  not  given  by  (3).  For  this  case  we  have  the  following 
theorem. 

Theorem  III.  If  the  analytic  function  f{z)  has  a  pole  at  z  =  00, 
then  the  residue  of  f{z)  at  that  point  is  the  negative  of  the  coefficient  of 
z~^  in  the  expansion  of  f(z)  for  values  of  z  in  the  neighborhood  of  z  =  00. 

Putting  z  =ip,  we  denote  the  transformed  function  by  <f>(z'). 

As  z  =  00  is  a  pole,  say  of  order  k,  of  the  given  function  /(z)  then 
2'  =  0  is  a  pole  of  the  same  order  of  «^(z').  Expanding  <p{z')  in  a 
Laurent  series  for  values  in  the  neighborhood  of  the  origin  we  have 

<t>i^')  =  ^*  +  ?SS+  ■  •  •  +  V^+  «o+ai2'  +  •  •  •  +anz'»+  •  •  •  .     (5) 

Replacing  z'  by  - ,  we  have  the  expansion  of  /(z)  in  the  neighborhood 
z 

of  2  =  00,  namely: 

/(2)=a_*2*+a_*+iz*-i+  •  .  •  +a_iZ+ao4-v+  '  '  '  +^+  •  •  •  . 

z  z 

=  a-*2*  +  a-A+iZ*-i  +   •  •  •  +  a-iz  +  00  +  ^  +^(^)'  (6) 

z 


Art.  53.]  RESIDUES  287 

where  z'^F{z)  is  holomorphic  in  the  neighborhood  of  z  =  oo.  In 
determining  the  residue  oi  j{z)  at  the  point  0=00  the  integral  defin- 
ing a  residue  is  to  be  taken  around  an  arbitrarily  large  circle  C  about 
the  origin  in  a  clockwise  direction.  The  resulting  integral  is  then 
the  negative  of  the  integral  taken  around  C  in  a  positive  or  counter- 
clockwise direction.     The  integral     /  F{z)dz  vanishes  by  Theorem 

X,  Art.  51.  We  have  then  from  (6)  by  aid  of  the  relations  given  in 
(2),  it  being  understood  that  the  integral  is  taken  in  a  clockwise 
direction  around  C, 

f{z)  dz  =  2  idai, 


or 


X 

^Jj{z)dz=-a,. 


2- 

Consequently,  from  the  definition  of  a  residue,  it  follows  that  the 
residue  at  00  of  an  analytic  function  having  a  pole  at  infinity  is  the 
negative  of  the  coefficient  of  z~'^  in  the  expansion  of  the  function  in 
the  neighborhood  of  the  point  z  =  00 ,  as  stated  in  the  theorem. 

It  is  of  interest  to  observe  that  a  function  can  have  the  point  z  =  qo 
as  a  regular  point  and  still  have  a  residue  at  that  point  different  from 
zero.     For  example,  the  function 

/(z)=ao  +  f 

is  holomorphic  in  the  neighborhood  of  z  =  00,  yet  it  has  a  residue 
—  a\  at  that  point. 

The  theory  of  residues  is  of  value  in  the  discussion  of  some  of 
the  important  properties  of  analytic  functions,  as  we  shall  see  in  the 
succeeding  articles.  It  may  also  be  applied  with  advantage  to  the 
evaluation  of  certain  integrals  of  functions  of  a  real  variable,  as  we 
shall  now  show.  First  of  all,  we  shall  show  how  we  may  employ  the 
results  of  our  discussion  of  residues  to  evaluate  an  integral  of  the 
form 


I 


f{x)  dx, 


where  f{x)  is  the  quotient  of  two  polynomials.  In  order  that  this 
integral  shall  have  a  significance,  we  make  the  assumption  that  the 
denominator  is  of  degree  at  least  two  higher  than  the  numerator.* 
For  the  sake  of  simplicity,  we  shall  also  assume  that  the  denominator 
has  no  real  roots. 

*  See  Pierpont,  Theory  of  Functions  of  Real  Variables,  Vol.  I,  Art.  635. 


288  SINGLE-VALUED   FUNCTIONS  [Chap.  VII. 

Let  us  consider  then  a  function  /(z)  which  is  holomorphic  along  the 
axis  of  reals,  and  with  the  exception  of  at  most  a  finite  number  n 
of  poles,  holomorphic  in  the  finite  upper  half  of  the  complex  plane. 
Consider  now  a  region  S  bounded  by  a  curve  consisting  of  a  seg- 
ment of  the  axis  of  reals  and  a  semicircle  C  about  the  origin,  lying 
in  the  upper  half  plane  and  having  the  radius  p.  We  select  the  value 
of  p  so  that  the  poles  of  J{z)  already  mentioned  shall  all  lie  in  S. 
Denoting  the  residue  of  the  given  function  at  each  pole  Zk  by  Rk, 
then  by  Theorem  II,  we  have  upon  integrating  about  the  contour 

j'  f(z)  dz  +  j  f{z)  dz  =  2inXRk,  (7) 

where    /     denotes  the  integral  along  the  axis  of  reals  between  —  p 

and  p. 

We  shall  first  consider  the  limit 


p  =  ao  t/C 


dz. 


dz 
Put  z  =  pe»*  and  hence  —  =  idd.     We  have,  therefore, 

z 

j  f{z)dz  =  i  jzfiz)de. 
But,  from  5,  Art.  17,  we  have 


I  ffiz)dz\=\  rzf(z)dd\^  rizfiz) 

\*Jc  1  «/o  I      «/o 


dB.  (8) 


Let  M  denote  the  upper  limit  of  |  z^{z)  \  upon  the  semicircle  C.    From 
(8),  we  have  then 

I  /  /(z)  dz   <M  r  dB  =  tM. 
I  Jc  Jo 

If  as  p  increases  without  limit,  M  approaches  zero,  we  have 


0  =  00   U  C 


z)  dz  =  0.  (9) 

We  now  regard  f{z)  as  the  quotient  of  two  polynomials,  where  the 
denominator  has  no  real  roots  and  is  of  degree  at  least  two  higher 
than  that  of  the  numerator.  Then  the  required  conditions  are 
satisfied,  for  f(z)  has  a  zero  of  at  least  order  two  at  infinity  and 

L  M  =  0; 

that  is,  the  result  expressed  by  (9)  follows. 


Art.  53.]  RESIDUES  289 

Moreover,  under  the  conditions  set  forth  in  the  statement  of  the 
problem  each  of  the  Umits 

L     f''f{x)dx,  L     f~''f{x)dx 

exists,  and  by  passing  to  the  limit  we  obtain  from  (7)  the  value  of 
the  integral   /      f(z)  dz,  namely 

1/    —  00 


fj  — c 


f{z)dz  =  2Tri^R 


k='l 


Ex.  1.     Evaluate  the  in 


Xoo 
-0 


dx 


-00   (X2  +  1)3 

The  function  /(z)  =  .  ^         ,3  has  a  pole  at  z  =  i.     Expanding  /(a)  in  powers 

(.2   +  i-j  ^ 

of  (z  —  i)  we  have  ,      \     j 

,, ,  _  _j 3 zi  n L-  1  ,--- 

^^^'  ~  8  (z  -  iy      16  (2  -  i)2      16  (z  -  i)  "^  ■  ■  ■  •  V  (^'»3^\^^^ 

Hence,  the  residue  of  /(z)  at  z  =  i  is  —  t^  •  ^ 

We  have  then 


/: 


dx       _      3  i    „    .      3 : 

00  (x2  +  1)3  ~  ~  16 


27rt   = 


The  method  of  residues  may  be  applied  also  in  evaluating  certain 
integrals  of  trigonometric  functions  when  the  integrals  are  taken 
between  the  limits  0  and  2  tt,  as  the  following  example  will  illustrate. 

do 


Ex.  2.     Evaluate  the  integral   i 


0     2  +  cos  d 
Putting  z  =  e»«,  we  have 

.       z  +  ^- 
,„      dz  ^      e'«  +  e-'«         ^  z 

de  =  ^,      cos»  =  — 2 —  =  -2~' 

whence  we  obtain 

2-^       d9  I  iz  ,       2   r  dz 


/-Zt       do       ^    I  iz  .   ^  2   /• 

Jo     2  +  cosff        I    1/     ,  .    ,  1\   ^      *■  Jc 
JcH^  +  ^  +  zj 


C22  +  4Z  +  1 

■^; 

The  integrand  /(z)  has  poles  at  the  points 

z  =  -2±V3. 
The  path  C  is  determined  by  the  substitution  z  =  e»*.     As  fl  varies  from  0  to 
2  TT,  z  describes  the  unit  circle  about  the  origin.     Of  the  two  poles  only  one,  namely, 
2  =  —  2  -f  V3,  falls  within  C.     The  residue  of  /(z)  at  this  pole  is  found  to  be 
1 


2  V3 


Hence  we  have 


Jo     2  +  cos «       t  2VS        3 


\ 


290  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

64.  Rational  functions.  Fundamental  theorem  of  algebra.  We 
have  defined  a  rational  integral  function  of  2  as  a  function  of  the 
form 

G{z)  =  oo  +  aiZ  +   •  •  •   +  an2", 

and  a  rational  fractional  function  as  the  quotient  of  two  such  func- 
tions having  no  common  factor;  that  is, 

f(   \   —  ^l(^)   _  Qq  +  Oil2  4-    •    •    •    +  Qing"  /.v 

•'^^^      G,{z)       /3o  +  /3i2+  .  .  •  +i3„2-'  ^'^ 

where  if  n  =  m,  then  the  common  value  of  n  and  m  is  said  to  be  the 
degree  of  /(z) ;  otherwise  the  larger  of  the  two  numbers  is  the  degree. 
Every  single-valued  algebraic  function  is  necessarily  a  rational 
function.  For,  if  w  is  an  algebraic  function  of  z,  then  the  two  are 
connected  by  the  irreducible  equation 

/o(2)W+/l(2)w;"-'+    •   •    •    +/n(2)  =  0, 

where/o(2), /i(2),  .  .  .  , /„(z)  are  integral  rational  functions.  If«;is 
single-valued,  this  equation  must  be  linear,  and  hence  solving  for  w, 
we  have 

__SM 
"^       Mz)' 

that  is,  ly  is  a  rational  function  of  2. 

We  shall  now  see  how  the  two  classes  of  rational  functions  are 
uniquely  determined  by  the  character  of  their  singularities.  As  we 
have  seen,  every  analytic  function  that  is  not  a  constant  must  have 
at  least  one  singular  point.  For  rational  integral  functions  we  may 
state  the  following  theorem. 

Theorem  I.  The  necessary  and  sufficient  conditions  that  a  single- 
valued  analytic  function  is  a  rational  integral  function  are  that  it  has 
no  singular  point  in  the  finite  portion  of  the  complex  plane  and  that  it 
has  a  pole  at  infinity. 

We  shall  first  show  that  the  given  conditions  are  necessary.  Let 
/(z)  be  a  rational  integral  function;  that  is,  let 

/(z)  =  ao  +  ociZ  +  azZ^  -}-•••+  an2", 

where  n  is  an  integer.  This  function  is  holomorphic  for  all  finite 
values  of  z,  and  hence  has  no  singular  points  in  the  finite  region  of 
the  complex  plane.     To  determine  the  nature  of  the  function  for 


Abt.  54.]  FUNDAMENTAL  THEOREM  OF  ALGEBRA  291 

z  =  00,  we  put  z  =  —,  and  examine  the  transformed  function  <^(2') 
for  z'  =  0.     We  have 

which  shows  that  <i>{z')  has  a  pole  of  order  n  at  the  origin,  and  hence 
j{z)  has  a  pole  of  the  same  order  at  the  point  z  =  ao. 

The  given  conditions  are  also  sufl&cient.     To  show  this  we  assume 
that/(2)  has  no  singular  point  of  any  kind  in  the  finite  region  and  that 
it  has  a  pole,  say  of  order  n,  Sit  z  =  cc.     As  we  have  seen  in  Art.  52 __^..^ 
the  expansion  of  /(z)  for  values  of  z  in  the  neighborhood  of  infinity 
is  then  of  the  form  V^^ 

f{z)  =  anZ""  +  an-iz''-^  +   •  •  •  +  ai2  +  F{z), 

1  where  F{z)  has  z  =  oo  as  a  regular  point.  Since  /(z)  is  holomorphic 
everywhere  in  the  finite  portion  of  the  plane,  it  follows  that  the 
^  same  expansion  holds  for  all  finite  values  of  z  and  that  F{z)  must  be 
holomorphic  in  the  finite  portion  of  the  plane  as  well  as  in  the 
neighborhood  of  infinity.  The  function  F{z)  is  therefore  a  constant 
(Theorem  XII,  Art.  51) .    Denoting  this  constant  by  oo,  we  may  write 

/(z)  =  anZ""  +  a„_i  z"-i  +  •  •  •  +  aiZ  +  oo, 

and  hence  /(z)  is  a  rational  integral  function  as  the  theorem  requires. 

From  this  discussion  it  follows  at  once  that  a  rational  integral 
function  is  fully  determined,  except  as  to  an  additive  constant,  when 
the  principal  part  of  the  expansion  for  the  pole  at  infinity  is  known. 

We  can  now  establish  the  fundamental  theorem  of  algebra,  which 
may  be  stated  as  follows: 

Theorem  II.     i//(z)  *s  a  rational  integral  function,  then  the  equation 
f{z)  =  ao  +  aiZ  +  aaz^  +   •  •  •   +  ccnZ^  =  0  (2) 

ha^  at  lea^t  one  root. 

By  Theorem  I  /(z)  has  no  singular  points  in  the  finite  portion  of 
the  plane  and  has  a  pole  at  infinity.     Putting  z  =  — ,  it  follows  that 

0(2')  =  f  {-\  has  a  pole  at  z'  =  0.     By  Theorem  III,  Art.  51,  — r-^ 
is  then  holomorphic  in  the  neighborhood  of  the  origin.     Conse- 
quently, 7^  must  be  holomorphic  in  the  neighborhood  of  z  =  00. 
But,  as  we  have  seen,  every  analytic  function  which  is  not  a  constant 


292  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

must  have  at  least  one  singular  point  either  in  the  finite  region  or 
at  infinity.  Since  ttt  can  not  have  a  singular  point  at  infinity,  there 
must  be  at  least  one  point  in  the  finite  region,  say  Zo,  at  which  the 
function  77-v  has  a  singularity.  This  singular  point  can  not  be  an 
essential  singular  point  for  in  that  case  Zo  would  not  be  a  regular 
point  of  f{z).     Hence  2©  must  be  a  pole  of  7^:  and  consequently  a 

zero  point  of  the  given  function  f(z).     This  establishes  the  theorem. 

As  a  consequence  of  this  theorem,  it  may  be  shown  by  the  methods 
of  elementary  algebra  that  a  rational  integral  function  f{z)  may  be 
written  as  the  product  of  a  constant  times  n  linear  factors,  where 
n  is  the  degree  of  f(z) .  Consequently  the  equation  (2)  has  n  roots 
real  or  complex,  and  no  more,  each  root  being  counted  a  number  of 
times  equal  to  its  multiplicity. 

We  have  the  following  condition  that  an  analytic  function  is  a 
rational  fractional  function. 

Theorem  III.  The  necessary  and  sufficient  conditions  that  a  single- 
valued  analytic  function  is  a  rational  fractional  function  are  that  it  has 
at  most  a  pole  at  infinity  and  that  it  has  a  finite  number  of  poles  hut 
no  essential  singular  points  in  the  finite  portion  of  the  plane. 

The  given  conditions  are  necessary;  for,  if  f{z)  is  a  rational  fractional 
function,  it  can  be  written  in  the  form  given  in  (1),  where  ^2(2)  is 
not  a  constant.  The  finite  singular  points  of  f{z)  are  the  finite  singu- 
lar points  of  Gi(z)  and  the  finite  singular  points  of  przr  >  since  there 

is  by  hypothesis  no  factor  common  to  the  two.  By  Theorem  I 
Gi{z)  has  no  singular  point  in  the  finite  portion  of  the  plane.  By 
Theorem  II  ^2(2)  has  at  least  one  zero  point  in  the  finite  portion  of 

the  plane.  Consequently,  /,  ,  ,  has  at  least  one  pole  in  the  finite 
portion  of  the  plane.  Since  Gziz)  can  not  have  more  than  m  zero 
points,  777-r  can  not  have  more  than  m  poles.     Except  in  the  neigh- 

yjr2{Z) 

borhood  of  the  special  points  just  mentioned  77-7-^  is  holomorphic: 

ij2{Z) 

for,  in  the  neighborhood  of  every  other  finite  point  Gziz)  is  holomorphic 
and  different  from  zero.  It  follows  then  that  in  the  finite  portion 
of  the  plane  f{z)  has  no  essential  singular  points. 


Art.  54.]  RATIONAL  FUNCTIONS  293 

To  deteraiine  the  nature  of  the  function  J{z)  for  2  =  00,  we  put 
2  =  —  and  have  ..  ^ 

Consequently  <i>{z')  has  at  z'  =  0,  and  therefore  f{z)  has  at  2  =  00, 
a  pole,  a  regular  point  which  is  not  a  zero  point,  or  a  zero  point 
according  as  n  >  m,  n  —  m,  n  <  m. 

The  given  conditions  are  also  sufficient.  To  show  this  let  us  assume 
that  the  poles  of  f(z)  in  the  finite  region  are  at  /3i,  ^2,  .  .  .  ,  /3m,  and 
let  us  denote  their  orders  by  ki,  ki,  .  .  .  ,  km,  respectively.  Then 
from  the  definition  of  a  pole,  we  have 

Gi(2)    =   (2  -  ^{)HZ  -  /32)*^    ...    (2  -  ^m^-Kz),  (3) 

where  Gi{z)  is  holomorphic  over  the  entire  finite  region  and  is  differ- 
ent from  zero  for  2  =  /3i,  /32,  •  .  .  ,  /3m.  However,  if  Gi{z)  is  not  a 
constant,  it  must  have  at  least  one  singular  point,  and  since  this 
can  not  be  a  finite  point,  it  must  be  the  point  at  infinity.  More- 
over, this  singular  point  must  be  a  pole,  for  otherwise  f(z)  would  also 
have  an  essential  singularity  at  infinity.  It  follows  from  Theorem  I 
that  Gi(z)  is  either  a  constant  or  a  rational  integral  function. 
From  (3)  we  have 

...  ^ GA^ Grizl 

^^^^       (2  -  ^OHz  -  ^2)*^  ...  (2  -  /3,„)*".  ~  G2(2) ' 

where  ^1(2)  and  6*2(2)  have  no  common  factor.  Consequently,  f(z) 
is  a  rational  function  as  stated  in  the  theorem. 

A  single-valued  function /(2)  having  a  finite  number  of  poles  but  no 
essential  singular  points  in  a  given  region  S  is  said  to  be  meromorphic 
in  S.  Thus  a  rational  fractional  function  is  meromorphic  in  the  en- 
tire complex  plane.  The  function  w  =  tan  2  is  meromorphic  through- 
out the  finite  part  of  the  plane,  the  poles  being  the  zeros  of  cos  2. 

As  has  been  pointed  out,  a  rational  function  may  be  either  integral 
or  fractional.  We  may  now  combine  the  results  of  Theorems  I  and 
III  into  one  theorem  for  the  unique  characterization  of  rational 
functions.     That  theorem  may  be  stated  as  follows: 

Theorem  IV.  The  necessary  and  sufficient  condition  that  a  single- 
valued  analytic  function  is  a  rational  function  is  that  it  has  no  essential 
singularities. 

We  can  now  establish  the  following  theorem. 


294  SINGLE-VALUED   FUNCTIONS  [Chap.  VII. 

Theorem  V.  A  rational  function  is  definitely  determined,  except 
for  a  constant  factor,  by  its  zero  points  and  its  poles  in  the  finite  portion 
of  the  plane. 

Let  f{z)  be  the  given  rational  function.  There  can  be  but  a  finite 
number  of  zero  points  and  a  finite  number  of  poles.  Let  the  poles 
in  the  finite  region  be  /3i,  /32,  .  .  .  ,  j8m  having  respectively  the  orders 
ki,  ki,  .  .  .  ,  km-  These  points  must  then  appear  as  the  zero  points 
of  the  rational  integral  function  G2{z)  in  the  denominator  of  the  given 
function.     We  may  then  write 

G2(Z)   =  C2  (0  -  /3i)*<2  -  ^2)*.    ...    (0  -  ^„)*".. 

The  finite  zero  points  of  f{z)  appear  as  the  zero  points  of  the  rational 
integral  function  Gi{z)  of  f{z).  If  we  denote  these  zero  points  by 
ai,  Q!2,  .  .  .  ,  a„  and  their  orders  respectively  by  ri,  r^,  .  .  .  ,  r„, 
we  have 

Giiz)  =  Ci{z-  aiYiz  -  a^y^  .  .  .   {z-  a„)^». 

We  have  then 

•'^   ^  G^iz)  "^   (2  -  ^x)*<0  -  ^2)*^    .    .    .    {Z-  fimY-'  ^    ^ 

where  C  =  y^.    In  case  Giiz)  reduces  to  a  constant,  the  function 
C2 

f{z)  is  a  rational  integral  function  having  but  one  pole,  namely  z  =<xi. 

Any  rational  function  f{z)  can  also  be  expressed  as  a  constant  or  a 

rational  integral  function  plus  a  finite  number  of  rational  fractions 

of  the  form 

a 

where  a,  /3  are  real  or  complex  constants  and  j'  is  a  positive  integer. 
To  express  a  given  rational  function /(«)  in  this  form  consider  the  prin- 
cipal parts  of  the  expansion  at  the  various  poles.  Let  the  poles  oif(z) 
in  the  finite  region  be  /3i,  ^2,  .  .  .  ,  /3m,  having  respectively  the 
orders  A;i,  /ca,  .  .  .  ,  km.  Let  the  principal  part  of  the  expansion  at  j8i 
be 

(2  -  ^1)*'  "^  (2  -  ^1)*'-^  "^   *  *  *   "^  (0  -  /3i)  " 

Subtracting  this  from/(0)  we  have  left  a  function  that  is  holomorphic 
in  the  entire  finite  portion  of  the  plane  except  at  ^2,  ...  ,  jSm. 
Proceeding  in  the  same  manner  with  the  remaining  poles  just  enu- 
merated, we  finally  have  a  function  that  is  holomorphic  in  the  entire 


Art.  54.]  RATIONAL  FUNCTIONS  295 

finite  plane  and  that  has  at  most  a  pole  at  infinity;  that  is,  we  have 
a  rational  integral  function,  say  of  the  form 

C  +  CiZ  -\-  C2Z^  +    •    '    ■    -\-  CnZ"". 

We  then  have 

/(z)   =  C  +  CiZ  +  dZ^  +    •    •    •    +  CnZ"" 

^  {z-  /30*'  '^  {z-  ^i)*'-^  "^  "  *  "^  (z  -  ^i) 


(z  -  ^2)*^  '    (z  -  ^2)*^-^  ^  ^  (z  -  ^2) 


,  C-A-,„  I  C-fc^+l  C_l 

~f"  r«  _  «  ^*,n  "•"  c^  _  «  ^tm-1  -t-  •  •  •  -r 


The  terms  in  the  first  row  of  this  expansion,  aside  from  the  constant 
term  c,  constitute  the  principal  part  of  the  expansion  of  /(z)  in  the 
neighborhood  of  infinity.  Consequently,  we  have  the  following 
theorem. 

Theorem  VI.  A  rational  function  is  definitely  determined  except 
for  an  additive  constant  by  the  principal  parts  of  its  expansion  at  its 
various  poles. 

The  following  examples  furnish  illustrations  of  Theorems  V  and  VI 
and  are  left  as  exercises  for  the  student. 

Ex.  1.  If  a  function  has  no  other  poles  than  simple  poles  at  2  =  1,  —1,  and 
its  only  zero  points  are  of  the  first  order  and  located  at  2  =  i,  —i,  show  that  it  ia 
necessarily  of  the  form 

Ex.  2.  Show  that  a  fimction  having  no  singularity  either  in  the  finite  part 
of  the  plane  or  at  infinity,  other  than  a  pole  at  the  origin  with  the  principal  part 


is  of  the  form 


3,2 

23^22' 

.,.       3,2,  2g  +  3   ,  ^ 


Theorem  VII.     The  sum  of  the  residues  of  a  rational  function  is 


zero. 


Let  f{z)  be  the  given  rational  function.  This  function  will  have 
no  essential  singular  points,  but  will  have  a  finite  number  of  poles, 
one  of  which  may  be  at  infinity.  Let  C  be  any  circle  in  the  finite 
region  not  inclosing  any  of  the  poles  of  /(z)  and  not  passing  through 


296  SINGLE-VALUED   FUNCTIONS  [Chap.  VII. 

any  such  points.     This  circle,  described  in  a  clockwise  direction,  may- 
be regarded  as  the  boundary  of  the  region  exterior  to  it,  that  is  the 
region  containing  all  of  the  poles  oi  f(z). 
The  value  of  the  integral 


hfj^^^"^' 


taken  in  a  clockwise  direction,  is  by  Theorem  II,  Art.  53,  equal 

to  the  sum  of  the  residues  of  f(z). 
The  value  of  the  integral  is  unchanged 
except  as  to  sign  if  it  be  taken  in 
the  counter-clockwise  direction  in- 
stead. But  since  f(z)  is  holomorphic 
within  and  along  C  this  latter  in- 
tegral vanishes.  Hence  taking  the 
integral  in  the  clockwise  direction  it 
is  zero  also.  It  follows  then  that  the 
sum  of  the  residues  of  f(z)  is  neces- 
sarily zero  as  stated  in  the  theorem. 
Some  of  the  general  properties  of 
single-valued  analytic  functions  that 
lead  to  special  properties  of  rational  functions  may  be  readily  de- 
duced from  a  consideration  of  the  logarithmic  derivative,  namely 

m 

that  is  the  quotient  of  the  first  derived  function  by  the  function  itself. 
If  the  point  z  =  Zo  is  a  regular  point  of  f{z)  and  if  /(zo)  is  different 
from  zero,  then  this  point  is  a  regular  point  of  the  logarithmic  deriv- 
ative. We  shall  see,  however,  that  if  f(z)  has  a  zero  point  at  Zo, 
the  logarithmic  derivative  is  not  holomorphic  in  the  neighborhood 
of  Zo.    We  have  in  fact  the  following  theorem. 

Theorem  VIII.  If  f{z)  is  holomorphic  in  the  neighborhood  of  the 
point  z  —  Zo  and  has  at  this  point  a  zero  point  of  order  k  then  the  loga- 
rithmic derivaiive  has  at  the  same  point  a  simple  pole  and  a  residue  k. 

The  given  function  f{z)  has  a  zero  point  at  26  of  the  order  k,  and 
hence  it  can  be  written  in  the  form 

fiz)  =  iz-  zomz),  (5) 


Art.  54.]  RA.TIONAL  FUNCTIONS  297 

where  0o  is  a  regular  point  of  <l){z)  and  this  function  is  different  from 
zero  for  z  =  Zq.    We  have,  therefore, 

r(z)  =h{z-  z,Y-'<i>{z)  +  {z-  ZoWiz), 
and  hence  obtain 

fiz)       z-zo'^  <f>{z)'  ^""^ 

The  function  ——-  is  holomorphic  in  the  neighborhood  of  Zq.    The 

Laurent  expansion  of  -frr  in  powers  of  {z  —  zo)  contains  but  one  term 

k 
having  a  negative  exponent,  namely  ;  the  rest  of  the  terms 

Z  —  Zo 

have  positive  exponents,  since  they  arise  from  the  expansion  of  the 

fb'(z^ 

holomorphic  function  ')! .    Hence,  the  residue  is  A;  and  the  point 

<t>{z) 

z  =  Zo  is  a,  pole  of  order  one,  as  required. 

Theorem  IX.    If  f{z)  has  a  pole  of  order  k  at  z  =  zo,  then  the  loga- 
rithmic derivative  has  at  the  same  point  a  simple  pole  and  a  residue  —k. 

We  may  write 

(z  -  ZoYfiz)  =  <l>{z), 

where  (j>(z\  is  holomorphic  in  the  neighborhood  of  the  point  z  =  Zo, 
and  is  different  from  zero  for  z  =  Zo. 
We  have  then  for  z  9^  Zo 

^(^'  =  5^' 


Hence,  we  obtain 


(z  -  ZoY' 

fj^^^^_  _fc_ 

f{z)       4>{z)       z-zo 

di(z\ 

Since  ^-ri-  is  holomorphic  in  the  neighborhood  of  2  =  2o,  the  expan- 
<i>(z) 

sion  of  the  second  member  of  the  foregoing  equation  in  powers  of 

{z  —  Zq)  contains  but  one  term  with  a  negative  exponent,  namely 

—k  fiz) 

,  and  hence  the  residue  of  the  quotient  77^ at  zo'is,  —k  and  the 


point  2  =  2o  is  a  pole  of  order  one,  as  stated  in  the  theorem. 

In  the  previous  article  we  have  shown  that  if  f{z)  is  holomorphic, 


wfr 


598  SINGLE-VALUED  FUNCTIONS  [Chap.  VIL 

'except  for  a  finite  number  of  poles,  in  a  closed  region  bounded  by  a 

«curve  C,  then  the  integral    I   f{z)  dz  \s,  2in  times  the  sum  of  the 

residues  at  the  poles  of  that  region.  If  we  now  apply  this  result 
in  evaluating  the  integral  of  the  logarithmic  derivative,  we  obtain 
the  following  theorem. 

1       p  f'(z) 
Theorem  X.     The  integral  - — ;  /  -v. ,  /  dz  taken  in  a  positive  sense 

2tiJcj{z) 

around  the  boundary  C  of  a  closed  region  in  which  the  rational  function 

f{z)  is  holomorphic  except  at  a  finite  number  of  poles  is  equal  to  the 

number  of  zero  points  of  f{z)  in  this  region  diminished  by  the  number  of 

poles,  each  zero  poird  and  each  pole  being  counted  a  number  of  times 

equal  to  its  order. 

It  should  be  noted  that  Theorems  VIII,  IX  and  X  apply  when 
2o  is  the  point  at  infinity  observing  of  course  the  convention  as  to  the 
direction  of  integration  about  the  point  at  infinity.  In  that  case 
the  given  function  may  be  written  in  the  form 

f{z)  =  zMz), 

where  <l>(z)  is  holomorphic  in  the  neighborhood  of  infinity  and  differ- 
ent from  zero  f or  2  =  oo .  If  X  =  A;  >  0,  f{z)  has  a  pole  of  order  k 
at  infinity,  and  if  X  =  —k,  f{z)  has  a  zero  point  of  order  k  at  infinity. 
We  have  then  for  the  logarithmic  derivative 

f'{z)  ^  }^-'<l>(z)  +  zV^(g) 
fiz)  zMz) 


z  "^  0(z) 
Theorem  III,  Art.  52 
<f>'{z) 


f(z) 
By  Theorem  III,  Art.  53,  the  residue  of  jt—  is  —X,  provided 


<t>{z) 


is  holomorphic  in  the  neighborhood  of  2  =  00 .     This  condi- 


tion is  easily  seen  to  be  satisfied  by  substituting  —  for  z  and  examin- 
ing the  transformed  function  in  the  neighborhood  of  z'  =  0. 
This  discussion  leads  to  the  following  theorem. 

Theorem  XI.  A  rational  function  is  just  as  often  zero  as  infinite, 
when  the  entire  complex  plane  including  the  point  at  infinity  is  considered, 
each  zero  point  and  each  pole  being  counted  a  number  of  times  equal  to 
its  order. 


Art.  54.1  RATIONAL  FUNCTIONS  299 

This  theorem  follows  at  once  from  Theorem  VII  concerning  the 
sum  of  the  residues  of  a  rational  function.  It  was  shown  that  this 
sum  is  necessarily  zero.  Hence,  from  Theorem  X  the  number  of 
zero  points  must  equal  the  number  of  poles,  each  being  counted  a 
number  of  times  equal  to  its  order. 

The  foregoing  theorem  admits  of  a  generalization  as  follows: 

Theorem  XII.  A  rational  function  of  degree  X  takes  any  given 
vahie,  real  or  complex,  exactly  X  times. 

If  /(z)  is  rational,  then  F{z)  =  f(z)  —  C  is  also  a  rational  function, 
where  C  is  any  constant.  By  Theorem  XI  the  function  F{z)  must 
be  as  often  zero  as  infinity.  Hence,  we  may  say  that  f{z)  takes 
any  arbitrary  value  C'as  often  as  F{z)  becomes  infinite.  We  shall 
now  show  that  F{z)  becomes  infinite  a  number  of  times  equal  to  the 
degree  of  the  function /( 2),  thus  establishing  the  theorem. 

The  degree  of  F{z)  is  the  same  as  that  of  f{z) .  The  function  F{z) 
may  be  written  in  the  form 

„.   ,  ao  +  Q!i2  +    •    •    •    +  ttnZ"  ,  n  o     _zA 

Po-r  PiZ-\-   •  •   •   +  PmZ'" 

where  the  degree  X  of  F(z)  is  the  larger  of  the  two  numbers  n,  m,  or 
equal  to  either  m  or  n  in  case  m  =  n.  In  any  case  F{z)  has  a  pole  at 
every  point  where  the  denominator  vanishes;  for,  by  hypothesis  the 
numerator  and  denominator  have  no  common  factor. 

For  n  =  m  =  \  there  are  X  poles  in  the  finite  portion  of  the  plane 
corresponding  to  the  X  zero  points  of  the  denominator.  We  can 
show  as  follows  that  the  point  z  =  00  is  a  regular  point  of  F{z). 

Putting  z  =  -,,  the  transformed  function  <f)(z')  is 
z 

CXi  ^\ 

_  «»  +  7  +    •   •   •   +  ^^,  ^  gpz^  +  arz'^-'  +   .   .   .   +  a^ 

^       ^^^^^^  A.^^       /3oz'^  +  /3iz'^-^+  •  •  .  +^x' 

OC-. 

which  is  —  for  z'  =  0.     Since  <f}(z')  is  holomorphic  in  the  neighbor- 

Px 
hood  of  the  origin,  the  point  z  =  00  is  a  regular  point  of  F(z).     In  this 
case  then  the  number  of  poles  of  F{z)  is  equal  to  the  degree  of  the 
function. 

For  n  <  m,  there  are  m  poles  of  F{z)  in  the  finite  region  corre- 
sponding to  the  m  zero  points  of  the  denominator.     In  this  case  also 


300  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

the  point  z  =  oo  is  a  regular  point  of  F(z),  in  fact  F{z)  has  a  zero 
point  at  infinity;  for,  we  have 

/3o  +  p+  •  •  •  +^ 

which  is  equal  to  zero  for  z'  =  0.  The  total  number  of  poles  of  F{z) 
is  therefore  m  =  X,  the  degree  of  F{z). 

If  we  have  n  >  m,  then  in  addition  to  the  m  poles  corresponding 
to  the  zero  points  of  the  denominator,  F{z)  has  a  pole  of  order  n  —  m 
at  infinity;  for,  we  have 

Po  +  p  +   •  •  •   +^ 

which  has  a  pole  of  order  n  —  m  at  z'  =  0.  Hence  F(z)  has  a  pole 
of  the  same  order  at  infinity.  In  this  case  therefore  the  totality  of 
poles  is  equal  to  n  =  X,  the  degree  of  F(z).     Hence  the  theorem. 

55.  Transcendental  functions.  As  we  have  seen,  an  analytic 
function  that  is  not  algebraic  is  a  transcendental  function.  As 
all  single-valued  algebraic  functions  are  necessarily  rational,  it 
follows  that  any  single-valued  analytic  function  that  is  not  rational 
is  transcendental,  and  hence  we  can  now  readily  identify  such  func- 
tions by  means  of  their  singularities.  A  single-valued  transcendental 
function  must  have  an  essential  singularity;  for,  otherwise  by 
Theorem  IV,  Art.  54,  it  would  be  a  rational  function.  If  a  single- 
valued  analytic  function  G{z)  has  no  singularity  in  the  finite  region 
and  has  an  essential  singularity  at  infinity,  it  is  called  a  transcendental 
integral  function  of  z.  The  expansion  of  such  a  function  in  a  Mac- 
laurin's  series  gives 

G{z)  =  ao  +  aiz -\- a^Z^ -\-   ■  ■  ■  -j-  a„0"  -}-•••,  (1) 

which  converges  for  all  finite  values  of  z.     If  in  (1)  we  replace  z  by 
,  we  have  a  transcendental  integral  function  of ,  namely 


z  -  zo'  °  2  -  zo 

(?(-!-)  =  ao  +  r-^+  •••  +^^^+  •••.      (2) 

Vz  -  V  (z  -  Zo)  ^  (z  -  Zo)" 

From  the  form  of  the  expansion  it  will  be  seen  that  this  function  has 
but  one  singular  point,  namely  an  essential  singular  point  at  z  =  Zq. 


Art.  55.]  TRANSCENDENTAL  FUNCTIONS  301 

Conversely,  if  a  single-valued  analytic  function  has  an  essential 
singular  point  at  z  =  Zq  and  has  no  other  singular  points,  then  it 
can  be  expanded  in  the  form  given  in  (2)  and  hence  is  a  transcendental 

integral  function  of 

Z  —  Zo 

As  we  have  seen,  a  power  series  may  be  integrated  or  differentiated 
term  by  term  for  values  of  the  variable  within  the  circle  of  con- 
vergence, and  the  resulting  power  series  has  the  same  circle 
of  convergence  as  the  given  series.  Consequently,  the  integral  or 
the  derived  function  of  a  transcendental  integral  function  G{z)  is 
represented  by  a  power  series  that  converges  for  all  finite  values  of 
z  and  hence  is  itself  a  transcendental  integral  function. 

The  following  functions  are  transcendental  integral  functions: 

e^  =  l+2;  +  |-j  +  3-j+  .  .  •  , 

^       ^       z' 
sin2  =  0-— ,-h^_-^+  •  •  •  , 

z^        ^        ^ 
cos2  =  l-2-j  +  j-j-g-,+  •  •  •  , 

sinh2  =  0  +  3-,  +  ^  +  ;^j+  •  .  ., 

^       ^        ^ 
cosh2=l  +  2l  +  4-|  +  6-|+  •  •  •  • 

A  transcendental  integral  function  differs  from  a  rational  integral 
function,  in  that  it  may  have  no  zero  points  or  it  may  have  an  infinite 
number  of  zero  points.     For  example,  the  function 

j{z)  =  e 

is  different  from  zero  for  all  finite  values  of  z.  In  fact  we  may  state 
the  following  theorem. 

Theorem.  Any  transcendental  integral  function  f{z)  having  no 
zero  points  may  he  written  in  the  form 

f{z)  =  e«(*), 
where  G{z)  is  an  integral  function. 

Since /(z)  is  a  transcendental  integral  function,/' (2)  is  also  a  trans- 
cendental integral  function,  and  as  f{z)  has  no  zero  points,  then 

F(i^  -  •'"(^) 


302  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

is  an  integral  function,  as  is  also  the  function  defined  by  the  integral 
I '  F{z)  dz.    We  have  then  the  integral  function  4>{z)  defined  by  the 


relation 


<i>{z)  =  J'f{z)  dz  =  y-j^^  dz  =  log/  {z)  -  l0g/(2<,). 

Puttmg  <t>{z)  +  log/(3o)  =  G{z), 

we  have  log/(z)  =  G{z), 

whence  Jiz)  =  e^^^^ 

where  G{z)  is  an  integral  function. 
The  function 

j{z)=R{z)e^(^\ 

where  R{z)  is  a  rational  integral  function  and  G{z)  is  an  integral 
function,  is  a  transcendental  integral  function  having  a  finite  number 
of  zero  points.  As  an  example  of  a  transcendental  integral  function 
having  an  infinite  number  of  zero  points,  we  have 

/(«)  =  sin  2, 
which  is  zero  at  the  points 

2  =   0,        db  X,        ±  2  TT,     .     .     .     ,     ±  /CTT,     .     .     .     . 

A  transcendental  integral  function  differs  from  a  rational  integral 
function  in  still  another  way.  As  the  rational  integral  function  has  a 
pole  at  infinity,  there  always  exists  a  circle  about  the  origin  such  that 
for  all  points  exterior  to  this  circle  we  have  |  f{z)  \  >  M,  where  M  is 
an  arbitrarily  large  number.  Since  a  transcendental  integral  func- 
tion has  an  essential  singularity  at  2  =  oo,  the  given  function  may  be 
made  to  approach  any  value  as  z  becomes  infinite.  Consequently, 
there  are  always  values  of  z  exterior  to  any  circle  however  large 
about  the  origin  for  which  |  /(z)  |  >  M  and  also  values  of  z  for  which 
I  f{z)  I  <  e,  where  e  is  an  arbitrarily  small  positive  number. 

A  transcendental  function  having  poles  but  no  essential  singular 
points  in  the  finite  portion  of  the  plane  is  called  a  transcendental 
fractional  function.  This  distinction  between  transcendental  inte- 
gral and  transcendental  fractional  functions  is  suggested  by  the 
corresponding  distinction  between  rational  integral  and  rational 
fractional  functions.  In  both  cases  a  function  is  called  integral 
when  it  has  but  one  singular  point  and  that  is  at  2  =  oo.  Likewise 
in  both  cases  we  call  a  function  fractional  if  it  has  poles  and  no  other 
singular  points  in  the  finite  region. 


Aht.  56.]  MITTAG-LEFFLER'S  THEOREM  303 

The  functions 

-: — ,        tan  3,        sec  2 
smz 

are  illustrations  of  transcendental  fractional  functions,  for  they  have 
an  essential  singular  point  at  infinity  and  poles  but  no  other  singular 
points  in  the  finite  region.  Each  may  be  written  as  the  quotient  of 
two  integral  functions.  It  will  be  shown  later  (Art.  57)  that  every 
transcendental  fractional  function  can  be  written  as  the  quotient  of 
two  integral  functions. 

We  have  pointed  out  that  any  rational  function  can  be  expressed 
as  the  sum  of  a  rational  integral  function  and  fractions  of  the  form 


(z  -  ZoY 


<t^'i 


It  is  not  difficult  in  that  case  to  set  up  the  function  when  the  prin- 
cipal part  of  the  expansion  is  known  for  each  of  the  various  singular 
points,  which  for  rational  functions  consist  of  a  finite  number  of 
poles.  The  corresponding  problem  for  transcendental  functions, 
namely,  the  problem  of  setting  up  a  function  with  arbitrarily  chosen 
singular  points  and  with  corresponding  arbitrary  principal  parts,  is 
much  more  difficult.  The  question  of  the  existence  of  an  analytic 
function  having  a  given  infinite  set  of  singular  points,  with  given 
principal  parts,  will  be  considered  in  the  following  article. 

56.  Mittag-Leffler*s  theorem.  Suppose  we  have  given  any  in- 
finite set  of  numbers  Zi,  Z2,  .  .  .  ,  Zk,  -  •  .  ,  all  different,  having  the 
property  that 

1^x1  =  1^1=    •••    =1^*1=    •••, 

and  suppose  that 

L   0*  =  cx). 

t=oo 

Mittag-Leffler  was  the  first  to  show*  that  there  always  exists  a 
single-valued  analytic  function  having  these  points  and  no  others  as 
singular  points,  with  given  principal  parts  of  the  form 


Gr 


(j4^>        /.  =  1,2,3,...,  (1) 

where  Gk  is  an  integral  function  of  —^ — ,  rational  or  transcendental. 

Z         Zk 

If,  as  in  rational  functions,  we  add  together  the  functions  (1)  we  have 
*  See  EncyJdopddie  d.  Math.  Wiss.,  Bd.  II2,  p.  80. 


304 


SINGLE-VALUED  FUNCTIONS 


[Chap.  VIL 


in  this  case  an  infinite  series.  Whether  the  function  defined  by  this 
series  is  everywhere  holomorphic  except  at  the  points  Zk  depends 
upon  the  nature  of  the  convergence  of  the  series.     Mittag-Leffler 

showed  that  by  associating  with  each  principal  part  Gk  ( 1  a 

\z  —  Zk/ 

suitably  chosen  polynomial  the  series  can  be  made  to  converge  uni- 
formly and  hence  define  a  function  having  the  desired  properties.  His 
theorem  may  be  stated  as  follows: 

Theorem.    Given  an  infinite  set  of  points 

Z\,  Z2f  Z3y    .    .    .    f    Zk,    .    .    .    f 


such  that 

0<\zr\^  \z,\^ 


=   \Zk 


,  L   Zfc  =  00. 


There  exists  a  single-valued  analytic  function  which  is  holomorphic  for 
all  finite  values  of  z  9^  Zk  and  which  has  an  arbitrarily  chosen  integral 

function  of  —— — ,  namely  Gk  i  —^ —  j,  as  the  principal  part  of  its 

pansion  in  the  neighborhood  of  Zk. 


ex- 


Let 


<ri  +  <r2  +   •  •  •   +  at  +   •  •  •  (2) 

«k    be  a  convergent  series  of  positive 
The  function 

A;  =  1, 2, 3,  .  .  , 


■*^'^c;\'^*  terms. 


G> 


G- J' 


Fia.  101. 


G, 


\Z  -  Zk) 


is  holomorphic  everywhere  in  the 
finite  region  except  for  z  =  Zk. 
It  can  be  expanded  in  a  Mac- 
laurin  series,  and  this  series  con- 
verges and  represents  the  func- 
tion for  all  values  of  z  within  a 
circle  C*  (Fig.  101)  about  the 
origin  as  a  center  and  having 
I  zjfc  I  as  a  radius.  Consequently, 
we  may  write 


oo^k  +  oiij^  +  a.2jk^^  + 


+  anjfi''  + 


(3) 


Within  and  upon  a  circle  Ck  about  the  origin  and  having  a  radius 
Pk  =  0  \  Zk  \,  0  <  6  <  \,  the  series  in  the  second  member  of  (3)  con- 


Art.  56.]  MITTAG-LEFFLER'S  THEOREM  305 

verges  uniformly.    Then  there  exists  an  integer  m*  such  that  for  z 
within  or  upon  Ct'  we  have 


5)      an.A2" 
n  =  m;i— 1 


<<^*;  (4) 


that  is,  putting  Fkiz)  equal  to  the  polynomial 

«o.i  +  axjkZ  +  oLijkZ^  +   •  •  •   +  a».i-2;fc2"*~* 

formed  by  taking  the  first  mi  —  1  terms  of  the  series  (3),  we  have, 
for  z  within  or  upon  Ct', 


«.C-^j-^.« 


<<r*.  (5) 


Denote  by  R  the  radius  of  any  circle  C  about  the  origin.  Then  in 
the  sequence  1,  2,  3,  .  .  .  there  can  be  found  an  integer,  say  r,  such 
that  we  have 

R<Q\Zr\. 

For  all  values  of  z  within  or  upon  the  circle  C,  z  must  also  lie  within 

C '    C 

and  hence  for  these  values  of  z  (4)  holds  for  fc  =  r.  For  values  of  z 
within  or  upon  the  circle  C  consider  the  series 

By  (5),  each  term  of  this  series  is  less  in  absolute  value  than  the  corre- 
ct 

sponding  term  an  of  the  convergent  series  of  positive  terms  2j  <''*• 

ik=r 

Hence,  by  Weierstrass'  test  for  uniform  convergence  (Theorem  I, 
Art.  45)  the  series  (6)  converges  absolutely  and  uniformly  within  and 
upon  the  circle  C  As  each  term  in  this  series  is  a  holomorphic 
function,  it  follows  that  the  series  defines  a  function  which  is  holo- 
morphic in  the  region  bounded  by  C. 
The  expression 


iT\l       \z  —  ZkJ 


(7) 

is  the  sum  of  a  finite  number  of  functions,  each  of  which  is  holomorphic 
in  the  region  bounded  by  C,  except  for  those  points  of  the  set  2i, 
22,  ...  ,  Zr-i  which  lie  within  or  upon  C.  Combining  (6)  and  (7), 
we  have  a  function 

FA')-%y{^)-P.(z)\,  (8) 


306  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

which  is  holomorphic  in  the  region  bounded  by  C,  except  for  these 
same  points.  But  since  C  is  any  circle  about  the  origin  it  follows 
that  the  function  Fi{z)  has  the  properties  desired,  and  the  theorem  is 
established. 

If  now  we  add  to  the  function  Fi(z)  any  integral  function  G{z), 
rational  or  transcendental,  we  obtain  a  more  general  function 

F{z)  =  F,{z)  +  G{z), 

which  also  has  the  same  finite  singular  points  with  the  same  principal 
parts  respectively  as  Fi(z),  and  consequently  F(z)  satisfies  the  condi- 
tions of  the  theorem.  Conversely,  if  F{z)  is  a  function  having  the 
singular  points 

2l,    Z2,    .    .    .    ,    Zjc,    .    .    . 

with  the  principal  parts 

\Z  -  ZiJ  \Z  -  ZiJ  \Z-  ZkJ 

respectively,  then 

F{z)  -  F,{z) 

is  an  integral  function  G{z),  and  therefore  we  have 

F{z)  =  F,{z)  +  G{z). 

For  special  cases  a  simpler  form  of  the  required  function  can  be 
shown  to  exist.  For  example,  let  us  consider  the  case  where  the 
function  is  to  have  simple  poles  at  the  points 

2l,    22,    ...    ,    Zkf    .    .    .    , 

and  where  at  each  pole  the  residue  is  one.  The  principal  part  of 
the  expansion  in  the  neighborhood  of  each  of  the  points  Zk  is  then 

The  series  (3)  then  becomes 

Z  —  Zk 

11 Z_  _  2" 

~  ,2  '    '    '  c>,.n+l  —     •    '    • 


Zk  Zk         Zk'-  Zk"" 

n+l' 


and  Pk{z)  =  -   y. 

The  series  (6)  can  then  be  written 


(9) 


But  we  have 


<  %  for  all  values  of  z  within  or  upon  the  circle  C, 


Art.  56.] 


MITTAG-LEFFLER'S  THEOREM 


307 


where  k  =  r.    Therefore,  for  fc  =  r  and  z  within  or  upon  C,  we  have 


X 


m^-l 


< 


1     Izl"'*-^ 

1  -  0 1  Zfc  h* ' 


(10) 


Hence,  from  (9)  we  have 

It  follows  that  the  series  of  holomorphic  functions  in  the  left-hand 
member  of  (9)  converges  uniformly  and  represents  a  holomorphic 

00 

function  within  and  upon  the  circle  C  if  the  series  ^ 


k=r 


converges 


uniformly.  By  the  Weierstrass  test  for  uniform  convergence  this 
series  converges  uniformly  if  its  terms  are  numerically  less  than  the 
corresponding  terms  of  a  convergent  series  of  positive  terms.  It  is 
sufficient  to  take  mu  —  h;  for,  since 


we  have 


k=r 


<d<l, 


wt* 


(11) 


*=r 


where  ]^  0*=  is  a  convergent  series. 

k=r 

If  there  exists  an  integer  p  independent  of  k  for  which  the  series 

1 


converges,  then  the  series 


*=i 


i=r 


(12) 


also  converges  and  we  may  put  nik  =  p.  In  this  case,  therefore,  the 
degree  of  the  polynomial  Pk(z)  does  not  need  to  increase  indefinitely 
with  k. 

Consequently,  if 

2l,  Z2,    .    .    .    ,    Zkj    .    .    . 

are  to  be  simple  poles  and  if  the  function  Fi{z)  has  the  residue  1 
at  each  pole,  then  from  (8)  Fi{z)  takes  the  simple  form 


Fx{z)  =  X 


1 


*=i 


Zk         Zk         Zk'' 


+ 


Zk' 


(13) 


308  SINGLE-VALUED   FUNCTIONS  [Chap.  VII. 

If  there  exists  a  constant  p  as  given  by  (12),  then  we  may  put  v  =  p. 
In  case  no  such  constant  exists,  then  we  need  at  most  put  *  v  =  k. 
In  Art.  44  it  was  shown  that  the  series 

converges.  By  aid  of  the  special  case  just  considered,  it  follows  that 
the  function 

has  simple  poles  of  residue  one  at  2  =  0  and  at  each  of  the  points 
2  =  fi.  With  the  exception  of  these  points  the  function  is  holo- 
morphic  in  the  entire  finite  plane. 

Differentiating  (14)  term  by  term  and  changing  the  sign  of  each 
term  we  have  another  important  function  namely 

By  differentiating  again,  we  have 

^''W—l-^X'cT^-  (16) 

The  three  functions  (14),  (15),  (16)  are  made  use  of  by  Weierstrass 
in  the  theory  of  elliptic  functions. 

57.  Expansion  of  functions  in  infinite  products.  In  addition  to 
expanding  an  analytic  function  by  means  of  an  infinite  series,  another 
method  is  often  employed,  namely  infinite  products.  Suppose  we 
have  the  infinite  sequence 

(1+ai),   (l  +  a2),   .   .   .   ,   (l-\-ak),   .... 
The  continued  product  of  the  first  n  elements  may  be  denoted  by 

Iln-na+a.).  (1) 

t=l 

If  at  most  a  finite  number  of  the  factors  (1  +  a^)  are  zero,  that  is  if 
there  exists  a  positive  integer  mi  such  that  for  A;  ,S  nii  all  of  the 
factors  are  different  from  zero,  then  the  product  (1)  is  said  to  con- 
verge if  the  limit 

L    11(1  + a.), 

*  Bore!  has  shown  that  it  is  suflScient  to  put  v  >  logk  (Lecons  sur  les  fonctions 
entikres,  p.  10). 


Abt.  67.1 


INFINITE  PRODUCTS 


309 


exists  and  is  different  from  zero.  If  the  product  tends  toward  zero, 
or  becomes  infinite,  or  if  for  any  reason  it  does  not  have  a  limit  as  n 
increases  indefinitely,  then  the  infinite  product  is  called  divergent. 

The  necessary  and  sufficient  condition  that  an  infinite  product 
converges  may  be  stated  as  follows: 

Theorem  I.  The  necessary  and  sufficient  condition  that  an  infinite 
product  converges  is  that  corresponding  to  an  arbitrary  positive  number 
6  there  exists  a  positive  integer  m  such  that  for  all  values  of  n  >  m  we 
have 

n+p 

n  (1  +  oc,)  - 1 


i=n+l 


<  6,  p  =  1,   2,       ... 


(2) 


Consider  the  sequence  of  the  products 

mi+l  OTi+2  n 

11'    n.        n- 


n  >  mi. 


(3) 


By  Theorem  VI,  Art.  12,  the  necessary  and  sufiicient  condition  that 
this  sequence  converges  is  that  for  every  positive  number  ei  there 
exists  an  integer  m  such  that 


n-\-p  n 

n-n 

mi  mi 


<  €i,       n>  m>  mi,       p  =  1,  2,  3,  . 


(4) 


We  shall  show  that  this  condition  is  equivalent  to  the  one  given  in 
the  foregoing  theorem.  If  the  given  infinite  product  converges,  then 
we  have 


There  then  exists  a  number  ilf  >  0,  such  that  for  all  values  of  n  >  Wi, 
we  have 

>  M.  (5) 


n 


Dividing  (4)  by  (5)  we  get 

n+p 

n 

w»i      -j 

ft 

n 

mi 


<^,        n>m,        p  =  l,2,3, • 


310 


SINGLE-VALUED  FUNCTIONS 


[Chap.  VI. 


Putting  ^  =  «,  this  result  may  be  written 


<  c,        n>  m,        p  =  1,  2, 3,  .  .  .  , 


n-1 

which  is  the  condition  given  in  the  theorem. 

Conversely,  suppose  we  have  given  the  inequality  (2).    We  may 
write 


n+P 

n 


n 


- 1 


n+p 

n 

w»i  1 


n 


But  from  the  condition  (2),  we  have  by  aid  of  the  above  relation 


n+p 

n 

m, 

-   1 

n 

n 

<  €,        n>  m,        p  =  1,  2,  3,  . 


(6) 


that  is,  we  have 


or 

n              (  n+p 

(1- 

on 

m, 

< 

11 

m, 

-€< 


<  (1  +  e) 


n+p 

n 


n 


n 


-Kt, 


,    n>m,    p  =  1, 2,  3, 


(7) 


Suppose  we  now  give  to  n  any  definite  value  greater  than  m,  say 
n  =  nil  -{-  V.    Since  e  may  be  chosen  arbitrarily,  it  follows  from  (7) 

n 

that  each  element  JJ  of  the  sequence  (3)  is  in  absolute  value  less 

than  a  positive  number  N,  which  is  the  largest  number  in  the  finite 
sequence 


m,+l 

mi+2 

mi+p-1 

11 

> 

11 

,    .    .    .    , 

11 

m, 

mi 

Wll 

,  (1  +  p) 


rrii+p 

n 


where  p  is  any  constant  greater  than  zero.  Likewise  each  element 
of  (3)  is  larger  in  absolute  value  than  a  positive  number  Ni,  which 
is  the  smallest  number  in  the  finite  sequence 


OTi+l 

m,+2 

mi+v—l 

11 

) 

11 

,      .      .      .      , 

n 

mj 

OTl 

m, 

,  (1-p) 


Tfli  +  V 

n 

mi 


Art.  57.J 


INFINITE  PRODUCTS 


311 


From  (6)  we  have 

n+p  n 

n-n 


<€ 


n 


n>  m,    p  =  1,2,3,  ...  . 


But  since  we  have 


n 


<  N,  it  follows  that  by  putting 


*=r 


where  e'  is  arbitrarily  small,  we  have 


n+p 

n 

11- 

mi 

11 

<c',        p  =  1,2,3,  .... 
Hence,  the  condition  given  in  (4)  is  satisfied  and  the  limit 

n 

i  n 


exists.     Suppose  the  limit  is  A.    Since 


n 


is  always  greater  than 


the  positive  number  Ni,  it  follows  that  A  9^  0.  Hence  the  given 
sequence  converges  and  the  demonstration  of  the  theorem  is  com- 
pleted. 

If  the  product  JJ  (1  +  ]  a*  |  )  converges,  then  the.  product 
JJ  (l+ai)  is  said  to  converge  absolutely.  If  an  infinite  product 
converges  but  does  not  converge  absolutely,  it  is  said  to  con- 
verge conditionally.  As  a  condition  for  absolute  convergence,  we 
have  the  following  theorem. 


Theorem  II.    //  the  series  ^  at  converges  absolutely,  then  the  in- 
finite  'product 

n(i+«*) 

converges  absolutely. 


k=\ 


From  the  convergence  of  the  series  Sat,  it  follows  that  only  a 
finite  number  of  the  factors  (1  +  a*)  can  be  zero.  We  assume  as 
before  that  for  fc  =  mi  the  factors  are  all  different  from  zero.  We 
may  then  write  for  n  >  mi 


TT   (1  _|_  aj,)   =  elog(l+a,„,)+log(H-o„,+i)  +  •  •  •  +log(l+o*)+  .  •  •  +log(l+a„).    ^g) 
i=mi 


312  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

The  given  product  will  then  converge  absolutely  if  the  series 
log(l+|a«.J)+log(l+|a..+,|)4----+log(l+|a*|)+  •  •  •  (9) 

convei^es.    But  this  series  converges,  if  the  series 

I  ai  1+  I  a2  I  +  •  •  •  +  I  a*  I  +  •  •  •  (10) 

converges;*  for,  the  ratio  of  the  general  terms  of  the  two  series, 
namely 

log(l  +  |«t|) 

has  the  limit  1  as  A;  becomes  infinite.  Since  the  series  (10)  con- 
verges by  hypothesis,  that  is  2  a*  converges  absolutely,  the  theorem 
follows. 

In  the  discussion  of  rational  integral  functions  we  saw  that  such 
functions  are  determined  except  as  to  a  constant  factor  when  the 
zero  points  are  known.  In  the  case  of  transcendental  integral 
functions  we  saw  that  the  given  function  might  have  no  zero  points, 
or  on  the  other  hand  it  might  have  an  infinite  number  of  such  points. 
If  we  have  given  an  infinite  set  of  points 

Zi,    Z2,    Z3,    •    '    •    ,    Zk,    '    •    •    , 

having  2  =  oo  as  a  limiting  point,  it  is  of  interest  to  see  whether  an 
integral  function  can  be  set  up  having  these  points  and  no  others  as 
zero  points.  Clearly  such  a  function  must  be  a  transcendental  func- 
tion, since  a  rational  integral  function  can  have  but  a  finite  number  of 
zero  points.  Weierstrass  has  shown  how  the  desired  function  can 
be  represented  by  an  infinite  product.  It  is  at  once  clear  that  if 
^{z)  is  such  a  function  and  G{z)  denotes  an  integral  function,  then 

F{z)  =  *(0)e<?(^) 

is  also  such  a  function,  since  as  we  have  seen  e^(*)  can  have  no  zero 
points.    The  function  e^^"^  plays  the  same  role  that  the  constant  factor 
does  in  the  representation  of  a  rational  integral  function  as  the  prod- 
uct of  a  finite  number  of  binomials. 
We  may  now  state  the  following  theorem. 

Theorem  III.    Given  a  set  of  points 

Zl,    Z2,    .    .    .    ,    Zk,    .    .    .  (11) 

not  including  the  origin  such  that 

\zi\  ^  \z2\  ^  ...   ^  \zk\  ^  ...,Lzfc  =  00. 

*  See  Bromwich,  Theory  of  Infinite  Series,  Art.  9. 


Art.  57.]  INFINITE  PRODUCTS  313 

There  exists  a  trq/nscendenial  integral  function  of  the  form 
$(2)  =  11(1  -  -]e^"'"2W  "•"•'■  "*";;ir^i  U 

having  the  points  Zk  and  no  others  as  zero  points.    Moreover,  the  function 

F(z)  =  e^^')  •  Hz) 
is  the  most  general  function  having  this  property. 

Consider  the  infinite  product 

n(i-l)/'©,  (12) 

where  </>ifc  ( —  lis  a  rational  integral  function  of  ( —  ]  as  yet  undetermined. 


The  factor 


Hy^^ 


is  called  a  primary  factor.     This  factor  has  one  and  only  one  zero, 

namely z  =  Zk.  Weshallshowhowthefunctions^jtf  —  J,  k  =  1,  2,  ...  .  , 

can  be  determined  so  that  the  infinite  product  (12)  will  converge 
in  an  arbitrarily  large  circle  and  define  a  function  having  the 
required  properties. 

Let  Ck  (Fig,  101)  be  the  circle  about  the  origin  as  a  center  having 
I  2ik  I  as  a  radius.  Let  Ck  be  a  circle  concentric  with  Ck  and  having 
the  radius 

Pk  =  e\zk\,      0  <  e  <  1. 

Denote  by  R  the  radius  of  any  circle  C  about  the  origin.  Then  there 
exists  an  integer  r  such  that  C  hes  within 

^r  ,    y^  r+1    ,    .    .    .    , 

and  that  within  and  upon  C  we  have  at  most  the  points 

Zl,    22,     .     .     .    Zr-l.  (13) 

The  product  of  the  corresponding  primary  factors  is  an  integral 
function  having  the  points  (13)  as  zero  points.  The  remaining 
factors  of  (12)  give  rise  to  the  product 

n(i  _£)/.©.  (14) 


314  SINGLE-VALUED  FUNCTIONS 

For  any  integer  q  we  have 


[Chap.  VII. 


nfi-^je  *^ =«*='•      *'   ^*^         (15) 


Hence,  the  convergence  of  the  infinite  product  (14)  will  follow  from 
the  convergence  of  the  series 

|j,„,(l_|)+,,(l)j.  (16) 

Since  all  of  the  points  z*,  k  ^r,  lie  outside  of  the  circle  C,  it  follows 
that  if  the  right-hand  member  of  (15)  converges,  it  defines  a  function 
that  does  not  vanish  for  z  within  or  upon  C.  Consequently,  the  same 
may  be  said  of  the  infinite  product  (14),  provided  this  product  is 
convergent. 

Expanding  log  (1 j  in  a  power  series,  we  have 

Suppose  we  let  <^*f  —  j  be  of  degree  w*  —  1,  where  m*  —  1  is  to  be  de- 
termined later.     We  may  then  put 

From  (16)  we  now  have 

We  have 

R  <e\zk\,        k  =  r. 

For  all  values  of  z  within  or  upon  the  circle  C,  we  have  then 

z 


(18) 


Zk 


<d,        k=r. 


By  aid  of  this  relation  -we  obtain 


1- 


»»* 


< 


1-d 


m^ 


Hence,  from  (18)  we  have 


(19) 


Art.  57.1  INFINITE  PRODUCTS  315 

It  follows  then  that  the  series  (14)  converges  uniformly  if  the  series 

00 

2 


k=r 


converges  uniformly,  which  it  does  if  nik  =  k,   as  we  have  seen 
(Art.  56). 

Each  term  of  the  series  (16)  is  holomorphic  for  values  of  z  within 
and  upon  the  circle  C,  and  since  the  convergence  is  uniform,  it  follows 
that  (16)  and  hence  (14)  defines  a  function  which  is  holomorphic 
within  and  upon  C.  But  C  is  any  circle  about  the  origin  as  a  center, 
hence  for  nik  equal  to  k  the  product 


n(.- 

i=i  \ 


defines  a  transcendental  integral  function  ^(z)  having  the  required 
properties.  If  we  desire  the  most  general  function  of  this  type,  all 
we  need  to  do  is  to  introduce  as  a  factor  the  most  general  function 
that  has  no  roots,  namely  the  function  e^^*\  where  G{z)  is  an  integral 
function. 

In  the  foregoing  discussion  the  origin  was  not  included  in  the  set 
of  zero  points.  If  it  is  desired  to  include  that  point  as  a  zero  point, 
say  of  the  order  X,  all  that  is  needed  is  to  add  the  factor  z^  and  write 
th/e  function 

F{z)  =e^^'^z^^z),  (20) 

which  is  also  a  transcendental  integral  function.  It  is  evident  that 
by  varying  the  function  G(z)  in  (20)  we  may  obtain  an  infinite  num- 
ber of  transcendental  integral  functions  having  the  points  given  in 
(11)  as  zero  points. 

We  have  seen  that  there  may  exist  an  integer  p  independent  of  k 
which  causes  the  series 

1 

Zk 

to  converge  and  hence  also  the  infinite  product  defining  $(2).  We 
may  then  put  lUk  =  p  and  have 

F{z)  =  e«(^)z^ IT (1  -  -- )e"* "^ 2W  "^  •  ■  ■  "^ P-iW     •  (21) 

it=i  \        2*/ 

In  the  discussion  of  integral  functions  it  is  desirable  to  introduce 
what  is  known  as  the  class  of  the  function.     For  this  purpose  let  us 


316  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

suppose  that  p  is  the  smallest  integer  that  makes  the  series  ^ 

converge.  Let  us  also  suppose  G{z)  to  be  a  polynomial,  say  of  degree 
g.  Then  the  class*  of  the  integral  function  F{z)  given  by  (21)  is 
defined  as  the  larger  of  the  two  integers  p  —  1  and  g.  Since  the 
degree  of  G{z)  can  be  changed  without  affecting  the  zero  points  of 
F{z),  we  may  so  choose  the  polynomial  G{z)  that  g  is  less  than  p  —  1 
and  hence  p  —  1  is  then  the  class  of  F{z).  The  class  of  any  rational 
integral  functions  is  zero,  as  is  also  that  of  the  transcendental  integral 
function 

/w=n(i-|} 

On  the  other  hand,  the  function 


-\R{'-t)' 


smz  =  z    II  1 1  —  —  je*' 


is  of  class  one. 

We  have  seen,  Art.  44,  for  example,  that  the  series 

converges.  Hence  there  exists  a  transcendental  integral  function  of 
the  form 

having  the  12  points  and  also  the  point  z  =  0  and  no  others  as  zero 
points.  This  function  is  the  (r-function  of  Weierstrass  and  is  of  great 
importance  in  his  development  of  the  theory  of  elliptic  functions. 
It  is  evidently  a  transcendental  integral  function  of  class  two. 

It  has  been  seen  that  every  rational  fractional  function  is  the 
quotient  of  two  rational  integral  functions.  Such  a  function  is 
uniquely  characterized  by  the  fact  that  it  has  at  most  a  pole  at 
2  =  00  and  only  a  finite  number  of  poles  in  the  finite  region  of  the 
complex  plane.  In  a  similar  manner,  we  have  defined  a  transcenden- 
tal fractional  function  as  one  that  has  an  essential  singularity  at 
z  =  00  and  only  poles  in  the  finite  region.  These  poles  may,  how- 
ever, be  dense  at  the  point  z  =  oc.  We  can  now  demonstrate  the 
following  theorem. 

*  The  term  doss  is  here  used  as  the  equivalent  of  the  French  word  genre  in- 
troduced by  Laguerre,  and  the  German  word  Hohe,  introduced  by  V.  Schaper. 
See  Borel,  Lemons  sur  les  fonctions  entihres,  p.  25;  also  Osgood,  Encyklopddie  d. 
Math.  Wi88.,  Vol.  IIj,  Part  I,  p.  79. 


Art.  58.]  PERIODIC   FUNCTIONS  317 

Theorem  IV.  Every  transcendental  fractional  juncticm  can  he  ex- 
pressed as  the  quotient  of  two  integral  functions. 

Let  the  points 

Zi,Z2,  .  .  .  ,  Zk,  .  .  .  ,  (22) 

where 

|2l|   ^    |22|=      •    •    •     ^    |2fc|   ^     •    •    •    ,    L     2,  =   00, 

be  the  poles  of  the  given  transcendental  function  f{z).  If  any  of 
the  poles  is  of  an  order  higher  than  one,  we  shall  regard  a  number  of 
the  points  Zk  equal  to  the  order  of  the  pole  as  coincident.  By  The- 
orem III  there  exists  a  transcendental  integral  function  Giiz)  having 
the  points  (22)  and  no  others  as  zero  points.  Moreover,  at  each 
point  the  order  of  the  zero  of  Giiz)  is  the  same'as  the  order  of  the  pole 
of  f{z) ;  for,  in  each  case  the  number  of  coincident  points  Zk  is  the 
same.  It  follows  that  the  product  Gi{z)  •  f{z)  has  no  singular  points 
in  the  finite  region.     Consequently,  we  have 

G2{z)fiz)  =  G,{z), 
where  G\{z)  is  an  integral  function.     It  follows  then  that 

as  the  theorem  requires. 

58.  Periodic  functions.  In  the  discussion  of  elementary  func- 
tions in  Chapter  IV,  attention  was  called  to  the  fact  that  certain  of 
those  functions  are  simply  periodic;  that  is,  the  function  remains 
invariant  under  a  translation  of  the  plane  by  means  of  the  relation 

z'  =  z-{-  no), 

where  n  is  an  integer  and  «  is  the  primitive  period  of  the  function. 
If  /(«)  is  the  given  function,  we  then  have 

fiz  +  no,)=f{z).  (1) 

In  the  illustrations  considered  the  period-strips  were  taken  parallel  to 
the  axes  of  coordinates.  It  is  not  necessary,  however,  to  choose 
the  strips  in  that  manner;  for,  if  we  locate  the  points 

2o  +  ncj, 
where  Zq  is  any  point,  and  draw  parallel  lines  through  these  points 
making  a  convenient  angle  different  from  zero  or  ^  with  the  X-axis, 
the  strips  bounded  by  these  lines  may  be  taken  as  period-strips.    As 


318  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

a  matter  of  fact,  the  boundary  lines  of  the  regions  of  periodicity  need 
not  even  be  straight  lines;  for,  all  that  is  essential  is  that  the  plane 
be  divided  into  congruent  strips  such  that  for  any  point  z  in  any  strip 
there  corresponds  a  point  2  +  wco  in  each  of  the  other  strips  for  which 
equation  (1)  holds.  It  is  of  importance  also  to  note  that  a  given 
function  may  repeat  its  values  in  a  period-strip;  for,  as  we  have  seen 
in  the  case  oi  w  =  cos  z,  the  period-strips  are  not  identical  with  the 
fundamental  regions  of  the  function. 

Functions  like  the  exponential  function  and  the  trigonometric  func- 
tions are,  as  we  have  seen,  simply  periodic.  Single-valued  analytic 
functions  may,  however,  have  two  independent  periods,  where  we 
understand  two  periods  of  a  function  to  be  independent  if  they  are 
not  integral  multiples  bf  the  same  primitive  period.  For  example, 
a  function  is  said  to  be  doubly  periodic  if  it  has  two  periods  2  coi,  2  W3, 
which  are  independent  of  each  other  and  of  z,  such  that  a  translation 
of  the  plane  by  either  of  the  relations 

2'  =  z  +  2  coi,  (2) 

2"  =  2  +  2  aJ3  (3) 

leaves  the  function  unchanged.    We  have  then  the  two  relations 

/(2-{-2coi)=/(2),  (3) 

/(2-t-2c03)=/(2).  (4) 

As  in  the  case  of  simply  periodic  functions,  any  translation  of  the 
complex  plane  by  means  of  the  relations 

2'  =  2  -f  2  mioji,         rwi  =  zt  1,     ±  2,  .  .  .  ,  (5) 

z"  =  2  +  2m3C03,        m3  =  db  1,     ±  2,  .  .  .  ,  (6) 

also  leaves  the  function  unchanged.  It  follows  at  once  that  any 
combination  of  the  translations  (5),  (6)  leaves  the  given  function 
invariant,  since  any  such  combination  may  be  regarded  as  a  suc- 
cession of  translations  by  means  of  (2),  (3).     We  may  then  write 

/(2  +  2  mioji  +  2  mscoa)  =  /(«),  (7) 

which  shows  that 

fi  =  2  miwi  -|-  2  m3W3  (8) 

is  likewise  a  period  of  the  given  function. 

If  all  of  the  periods  of  the  given  function  can  be  written  in  the  form 
(8),  that  is  if  every  such  period  can  be  expressed  as  the  sum  of  integral 
multiples  of  these  two  periods,  then  2  wi,  2  0)3  are  called  a  primitive 
period  pair. 


Art.  58.]  PERIODIC   FUNCTIONS  319 

In  order  that  any  two  pairs  other  than  2  wi,  2  W3,  for  example 
2  mi'coi  +  2  inz'o)3, 
2  w/'wi  +  2  mz'oiz, 
may  be  taken  as  a  primitive  pair,  it  is  sufficient  that  we  have 


mi     ms 
mi"  rriz 


=  ±1. 


(9) 


For,  let  2  wi',  2  ws'  be  any  two  independent  periods  of  f{z)  other  than 
2  oji,  2  ojs.     Putting 

2  m/  coi  +  2  ms'cos  =  2  w/, 

2  Wi"  coi  +  2  m3"co3  =  2  C03', 
and  solving  for  2  wi,  2  0)3,  we  have 

2m3V-2m3W          „           -2mi'W  +  2miW    .-.^x 
2coi  =  ^^ ,         2oj3=  ^ (10) 

If  A  =  ±1,  then  2  coi,  2  C03  are  each  a  sum  of  multiples  of  2  w/,  2  C03' 
and  consequently  any  period 

fl  =  2  miwi  -f  2  m3W3 
of  f{z)  can  be  written  in  the  form 

fi  =  2  nio)/  +  2  7130:3'; 
hence  2  oji',  2  W3'  are  a  primitive  period  pair. 

Theorem  I.    Let  f(z)  be  a  single-valued  doubly  periodic  analytic 
function  with  the  independent  periods  2  coi,  2  C03.     Then  if  f{z)  is  not 

a  constant,  the  ratio  —  can  not  be  real. 

COi 

Let  Us  assume  first  of  all  that  the  ratio  —  is  real  and  commen- 

COi 

surable,  say 

W3  _  p 
031      q' 

where  p,  q  are  integers  prime  to  each  other.  We  shall  show  that 
this  assumption  leads  to  a  conclusion  which  is  contrary  to  the  given 
hypothesis.  Since  2  coi,  2  03  are  periods,  it  follows  that  2  micoi  +  2  m3C03 
is  also  a  period.     But  from  the  assumed  relation,  we  may  write 

2miwi  +  2m3C03  =  2  coifmi  +  m3-)  =  2o33lmi-  +  ms) 

^  2  a?i(mig  +  msp)  ^  2  mjmiq  +  msp)        ..  .^ 
q  V 


320  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

Since  p,  q  are  relatively  prime,  mi  and  ms  can  be  so  chosen  that  * 

miq  +  map  =  1. 
Consequently,  we  obtain  from  (11) 

I    O^  2coi        2W3        rt 

2  miwi  +  2  macos  =  =  =  2  w, 

3  p 

where  2  w  is  a  period.    Hence,  we  have 

2  wi  =  2  gco,        2  W3  =  2  pco; 

that  is,  the  periods  2  wi,  2  ws  are  each  multiples  of  a  common  period 
2  w  and  hence  are  not  independent. 

Likewise  the  assumption  that  the  ratio  —  is  real  and  incommen- 

COl 

surable  leads  to  a  contradiction.     For,  let  —  be  converted  into  a  con- 

Wi 

tinued  fraction.     Then  —  must  lie  between  any  two  consecutive 

convergents,t  say 

Pk         Pk+i 

Qk  Qk+i 

Hence,  the  value  of  —  differs  from  either  of  these  convergents  by  less 

Wi 

than We  may  then  write 

qkqk+i 


W3  _  Pfc 

wi      qk 


<  1 


qkqk+i 

COi 


whence  I  qtois  —  pkosi  |  < 

qk+i 

But  since  qk+i  can  be  taken  as  large  as  we  please,  it  follows  that 
may  be  made  as  small  as  we  please.     From  the  foregoing  relation, 


qk+i 
it  follows  that  however  small  - — —  may  be,  values  of  mi  and  mz  exist 

Qk+l 

such  that  I  2  miwi  +  2  msm  |  =  1  12  |  is  numerically  arbitrarily  small. 
Consequently,  there  exists  a  set  of  values  z  dense  at  any  regular 
point  Zo  for  which /(z)  has  the  same  value /(zo).  This  condition  can 
not  exist  except  when  f(z)  is  a  constant. 

Since  the  ratio  —  can  be  neither  real  and  commensurable  nor  real 

and  incommensurable  it  must  be  complex. 

*  See  Chrystal,  Text-Book  of  Algebra,  Vol.  II,  p.  409. 
t  Ibid.,  p.  410. 


Akt.  58.]  PERIODIC  FUNCTIONS  321 

Let  2  wi,  2  a>3  be  a  primitive  period  pair  of  the  single-valued  doubly 

periodic  analytic  function  f{z).    Since  the  ratio  —  can  not  be  real, 

the  straight  line  joining  the  origin  with  2  wi  makes  an  angle  different 
from  zero  or  tt  with  the  line  joining  the  origin  with  2«3.  Conse- 
quently, the  set  of  complex  values 

12  =  2  rwiwi  +  2  W3«3 

is  represented  by  a  net  of  points  covering  the  complex  plane.  More- 
over, any  other  primitive  period  pair  of  f{z)  must  lead  to  the  same 
net.  If  00  is  any  point  in  the  region  of  existence  of  f{z),  then  in  the 
parallelogram 

2b,     2o  +  2  «l,    00  +  2  W3,    2!o  +  2  Wi  +  2  «3, 

the  given  function  takes  all  of  its  values.  This  parallelogram  is 
called  a  primitive  period-parallelogram.     If  we  put 

2  wi  4"  2  a>3  =  —  2  ci>2, 
we  have  the  relation 

2  Wi  -F  2  C02  +  2  0)3  =  0. 

By  drawing  parallel  straight  lines  through  the  points  of  the  net 
Zo  +  12  as  shown  in  Fig.  102,  we  have  a  set  of  congruent  period- 


FiG.  102. 

parallelograms  covering  the  entire  complex  plane.  If  2o  is  a  singular 
point,  then  each  point  2o  +  12  is  likewise  a  singular  point.  It  is 
often  convenient  to  choose  zq  so  that  no  singular  point  of  /(z)  lies 
upon  the  boundaiy  of  the  period-parallelograms.  This  choice  is 
always  possible  if  the  number  of  singular  points  of  f{z)  in  a  period- 
parallelogram  is  finite.  For  example,  if  the  singular  points  of  f{z) 
are  restricted  to  poles,  then  there  are  but  a  finite  number  of  poles  in 
the  initial  period-parallelogram,  and  hence  by  the  proper  choice  of 
Zo  none  of  these  poles  will  lie  upon  the  boundary  of  any  period-parallel- 
ogram. 


322  SINGLE-VALUED   FUNCTIONS  [Chap.  VII. 

We  shall  now  set  up  the  convention  that  the  points  on  one  pair  of 
adjacent  boundary  lines  of  a  primitive  period-parallelogram  belong 
to  the  parallelogram  and  that  the  points  on  the  other  two  boundary- 
lines  do  not.  For  example,  in  the  parallelogram  Pi,  Fig.  102,  the 
points  on  the  boundary  joining  zo,  Zo  +  2  wi  and  Zq,  Zq  +  2  cos,  the 
points  2o  +  2  oji,  2o  +  2  0)3  excepted,  are  considered  as  belonging  to 
the  period-parallelogram  but  the  points  on  the  other  two  boundary 
lines  do  not.  It  is  sufficient  to  study  the  behavior  of  a  doubly  peri- 
odic function  for  values  of  z  in  any  period-parallelogram,  just  as  in 
the  case  of  simply  periodic  functions  it  is  sufficient  to  examine  the 
function  for  values  of  z  in  any  one  of  the  various  period-strips.  This 
fact  simplifies  the  discussion  of  periodic  functions.  We  shall  use 
the  term  period-region  to  mean  either  a  period-strip  or  a  period- 
parallelogram  according  as  the  given  function  is  simply  or  doubly 
periodic. 

We  shall  have  no  occasion  to  discuss  in  this  connection  functions 
having  more  than  two  independent  periods,  for  it  may  be  shown  that 
a  single-valued  analytic  function  can  at  most  be  doubly  periodic* 

As  an  illustration  of  a  doubly  periodic  function,  let  us  consider  the 
function  p'{z)  of  Weierstrass,  which  is  defined  by  the  relation  (Art.  56) 

=  -2S(^-  (12) 

As  we  have  seen,  this  function  is  holomorphic  in  the  finite  region  except 
for  the  doubly  infinite  set  of  fi-points.  Replacing  z  by  z  ±  2  wi,  we 
have  from  (12) 

^-(zd=2.0  =  -2X(,^2cl.-Q)3 

=  _2V I 

^\z-  (fiT2«i)P 

From  an  examination  of  Fig.  102,  it  will  be  seen  that  the  set  of  points 
z  —  (12  T  2  wi)  is  the  same  as  the  set  of  points  z  —  12,  the  points 
being  taken  in  another  order.  The  order  of  the  terms  in  the  series 
defining  p'{z)  is  immaterial  since  the  series  converges  absolutely. 
Consequently,  we  have 

^'(z±2a;i)  =  p'(z). 

*  Forsyth,  Theory  of  Functions,  2d  Ed.,  p.  230. 


Art.  58.]  PERIODIC  FUNCTIONS  323 

Similarly,  it  may  be  shown  that 

S?'(2±2co3)  =  ^'(2). 

Hence,  it  follows  that  ip'{z)  remains  invariant  by  the  translation 

z'  =  z-\-2  micoi  +  2  mzuz, 

that  is, 

p\z  +  Q)  =  p\z), 

and  the  given  function  is  therefore  doubly  periodic. 

Theorem  II.  //  a  single-valued  periodic  analytic  function  f(z)  is 
holomorphic  in  any  given  period-region,  it  is  a  constant. 

As  we  have  seen,  a  periodic  function  takes  the  same  value  at  the 
corresponding  points  of  the  various  period-regions.  If  the  given 
function  is  holomorphic  in  any  period-region,  then  in  that  region  it  is 
continuous  and  bounded.  But  if  it  is  bounded  in  any  period-region, 
it  is  bounded  over  the  entire  complex  plane.  Hence  f{z)  has  no 
singular  point  and  is  therefore  a  constant  by  Theorem  XII,  Art.  51. 

We  have  also  the  following  theorem. 

Theorem  III.  A  single-valued  periodic  analytic  function  w  =  f(z) 
which  is  not  a  constant  is  necessarily  a  transcendental  function  of  z. 

By  Theorem  II  each  period-region  must  contain  at  least  one  singu- 
lar point.  In  the  case  of  doubly  periodic  functions  these  singular 
points  have  the  point  2  =  00  as  a  limiting  point,  and  consequently 
that  point  is  an  essential  singular  point.  The  same  result  follows  in 
the  case  of  simply-periodic  functions  if  singular  points  appear  in 
the  finite  portion  of  the  various  period-strips.  However,  the  point 
at  infinity  is  a  point  in  each  period-strip  and  this  point  may  be  the 
only  singular  point.  In  this  case  the  point  2  =  00  must  also  be  an 
essential  singular  point.  For,  if  2o  is  any  finite  point  in  one  of  the 
period-strips,  the  function  takes  then  the  same  finite  value  /(20)  at 
each  of  the  corresponding  points  Zq  -\-  ho,  where  «  is  the  primitive 
period  of  the  function  and  k  has  the  values  1,  2,  3,  ...  .  The 
points  have  the  limiting  point  2  =  00,  but  the  limit  of  the  function 
as  k  becomes  infinite  is  the  finite  value  /(20).  Since  we  obtain  by  a 
particular  approach  to  the  point  2  =  00  a  finite  limiting  value  of 
the  function,  that  point  can  not  be  a  pole.  Since  it  is  a  singular 
point,  it  must  then  be  an  essential  singular  point.  The  given  func- 
tion can  not  be  a  rational  function;  for,  in  that  case  the  point  2  =  00 


324  SINGLE-VALUED  FUNCTIONS  [Chap.  VII. 

can  not  be  an  essential  singular  point.  Since  all  single-valued  alge- 
braic functions  are  rational,  and  hence  can  have  no  other  singular 
points  than  poles,  the  given  function  must  be  transcendental. 

Theorem  IV.  Let  f(z)  be  a  single-valued  dovbly  'periodic  analytic 
function  having  only  a  finite  number  of  singular  points  in  each  period- 
parallelogram.  The  integral  j  f(z)  dz  taken  around  the  contour  of  a 
period-parallelogram  is  zero. 

Denote  the  contour  of  a  period-parallelogram  by  C.  If  Zo  is  one 
of  the  comer  points  of  the  parallelogram,  then  the  other  points  may- 
be taken  as  indicated  in  Fig.  103,  Zo  being  so  chosen  that  no  singular 


Fig.  103. 


points  lie  on  the  contour  of  the  period-parallelogram.     The  integral 
taken  in  a  positive  direction  around  the  contour  is 

f{z)dz=    /  f(z)dz-\-   /  f(z)dz 

f{z)dz+   /         f{z)dz. 

2o4-2a)i+2«|  t/2o+2Mj 

We  can  combine  the  first  and  third  integrals  in  the  right-hand  mem- 
ber, and  likewise  the  second  and  fourth,  thus  obtaining 

/»  /»«6+2«,  /%+2«, 

JJ{z)  dz^'J^         lf(z)-f(2^2c^)ldz-  J  mz)-f(z-\-2o}0ldz. 

But  as  f{z)  is  doubly  periodic  having  the  periods  2  coi,  2  wa,  we  have 

f(z  +  2<^)=f(z), 
f{z  +  2<^,)=f{z), 


Akt.  58.]  PERIODIC  FUNCTIONS  325 

from  which  it  follows  that  both  of  the  integrals  in  the  second  member 
of  the  foregoing  equation  vanish.     Hence,  we  have 


X- 


as  the  theorem  requires. 

By  aid  of  this  theorem  we  can  now  estabUsh  the  following  theorem. 

Theorem  V.  The  sum  of  the  residues  in  a  period-parallelogram  of 
a  single-valued  doubly  periodic  analytic  function  having  in  any  period- 
parallelogram  only  a  finite  number  of  singular  points  is  zero. 

The  residue  of  a  function  at  an  isolated  singular  point  was  defined 
as  the  value  of  the  integral 


isX^(^)'^' 


where  C  is  a  closed  path  inclosing  no  other  singular  point.  It  has 
also  been  shown  that  this  integral  taken  around  the  contour  of  a 
region  containing  a  finite  number  of  singular  points  is  the  sum  of  the 
residues  of  the  function  at  these  points.  It  follows  then  from  The- 
orem IV  that  if  a  function  f(z)  satisfies  the  stated  conditions,  the  sum 
of  its  residues  at  the  singular  points  in  a  period-parallelogram  must 
be  zero. 

Theorem  VI.  The  number  of  zero-points  in  any  period-parallel- 
ogram of  a  single-valued  doubly  periodic  analytic  function,  which  is  not 
a  constant  and  has  no  singular  points  other  than  poles,  is  equal  to  the 
number  of  poles  in  this  period-parallelogram,  each  zero  point  and  eaxih 
pole  being  taken  a  number  of  times  equal  to  its  multiplicity. 

Let  f(z)  be  the  given  function.  Since  it  is  analytic,  its  zero  points 
are  isolated,  and  hence  in  any  finite  region  there  can  only  be  a  finite 

f(z) 
number.     By  hypothesis  f(z)  is  periodic.     Hence,  f'(z)  and  ^p-  are 

Hkewise  both  periodic.     Moreover,  since /(«)  has  but  a  finite  number 

of  zero  points  in  any  finite  region,  jrr-r  can  have  but  a  finite  number 

of  poles  in  such  a  region.     Consequently,  the  quotient  -y^  has  but 

a  finite  number  of  poles  in  any  period-parallelogram.  By  Theorem 
V  we  have 


l-mjyfiz) 


326  SINGLE-VALUED   FUNCTIONS  [Chap.  VII. 

where  7  denotes  the  contour  of  any  period-parallelogram  so  chosen 

f  (z) 
that  no  poles  of  -^  or  of  f(z)  lie  upon  7.    But  by  Theorem  X, 

Art.  54,  it  follows  that  the  foregoing  integral  is  equal  to  the  number 
of  zero  points  of  f(z)  in  the  period-parallelogram  bounded  by  7,  minus 
the  number  of  poles  of  f{z)  in  the  same  parallelogram.  Since  this 
difference  is  zero,  we  have  the  given  theorem. 

The  foregoing  theorem  may  be  generalized  as  follows: 

Theorem  VII.  In  any  period-parallelogram  of  a  single-valued 
doubly  periodic  analytic  function  f{z),  which  is  not  a  constant  and  has 
in  the  finite  region  no  other  singular  points  than  poles,  takes  any  arbi- 
trary value  C  a  number  of  times  equal  to  the  number  of  its  poles,  each 
pole  being  taken  a  number  of  times  equal  to  its  multiplicity. 

To  prove  this  theorem  consider  the  function 

F(z)=f{z)-C. 
The  conditions  of  Theorem  VI  are  satisfied  by  F(z),  and  hence  the 
number  of  its  zero  points  is  equal  to  the  number  of  its  poles  in  any 
period-parallelogram.  But  the  poles  of  F{z)  are  at  the  same  time 
poles  of  /(z).  Moreover,  at  the  zero  points  of  F(z),  f{z)  takes  the 
value  C.  Hence,  the  given  function  f{z)  takes  the  arbitrary  value 
C  in  each  period-parallelogram  a  number  of  times  equal  to  the  num- 
ber of  its  poles  in  that  parallelogram. 

If,  as  in  Theorem  VI,  our  attention  is  confined  to  functions  having 
no  singular  points  in  the  finite  part  of  the  plane  except  poles,  it 
follows  that  there  must  be  at  least  one  pole  in  each  period-parallel- 
ogram. If  there  is  but  one  pole,  its  residue  must  be  zero;  conse- 
quently the  order  of  the  pole  must  be  at  least  two.  If  only  simple 
poles  appear,  each  period-parallelogram  must  contain  two  or  more 
such  poles  in  order  that  the  residue  may  be  zero.  Doubly  periodic 
functions  having  only  poles  in  the  finite  region  form  an  important 
class  of  functions  called  elliptic  functions.  We  shall,  however,  not 
consider  the  special  properties  of  such  functions  in  the  present 
volume. 

EXERCISES 

1.  Show  that  every  rational  integral  function  f{z)  is  uniquely  determined 
for  all  complex  values  of  z  as  soon  asf(x)  is  known  for  all  real  values  of  x  from  zero 
to  one. 

2.  Given  the  fimction 

■""'         2(2*  -Z-2) 


Art.  58.]  EXERCISES  327 

Represent  this  function  by  an  infinite  series  for  values  of  z;  (o)  within  the  unit 
circle  about  the  origin;  (6)  within  the  annular  region  between  the  concentric 
circles  about  the  origin  having  respectively  the  radii  1  and  2;  (c)  exterior  to  the 
circle  of  radius  2. 

3.  Can  sin  z  ever  be  greater  than  one?  If  so  at  what  points?  How  do  the 
zeros  of  sin  z  compare  with  those  of  sin  x,  where  x  is  real?  What  solutions  can 
the  equation  cos  z  =  1  have? 

4.  Given  the  infinite  series 

From  the  form  of  this  series,  how  do  we  know  that  it  defines  a  function  f{z)  that 
is  holomorphic  in  the  entire  finite  portion  of  the  complex  plane?  Knowing  the 
expansion  of  e^,  how  can  it  be  shown  that  /(z)  =  e^?  How  can  this  same  method 
be  used  to  obtain  the  expansion  of  sin  z,  cos  z  from  the  expansion  of  sin  x,  cos  x? 
6.   Given  the  function 

m  -  ^y 

Is  /(z)  an  analytic  function?  What  is  its  region  of  existence?  Find  a  region  S 
in  which  the  function  is  holomorphic.  Locate  the  singular  points  and  classify 
the  singularities  of  the  function  at  these  points.  Does  the  function  have  any 
zero  points? 

6.  Indicate  the  general  form  which  the  infinite  expansion  of  an  analytic 
function  /(z)  must  have  in  the  neighborhood  of  Zo,  if  /(z)  has  (a)  a  pole  of  order 
3  at  Zo,    (6)  a  zero  of  order  3  at  Zo,    (c)  an  essential  singularity  at  Zo. 

What  is  the  natiu-e  of  the  fxmction  F{z)  =  j^-^  at  Zo  in  each  case? 

7.  Locate  the  zero  points  of  sin  -•     What  is  the  nature  of  this  function  at  the 

Umiting  point  of  these  zeros?  From  this  conclusion  what  can  be  said  of  the  sin- 
gular points  of  CSC  zf    Why  is  esc  z  a  transcendental  fractional  function? 

8.  Given 

•^^^^  =  (z-l)(z-3)' 

If  this  function  is  expanded  in  a  Taylor  series  about  z  =  2,  how  large  can  the  circle 
of  convergence  be?  Expand  the  given  function  in  powers  of  z  and  determine  the 
circle  of  convergence.  i_^         ^_j_       _•  _L    \  -f/— ^  r-      -_t 

9.  GiveA  the  analytic  expression  ^_-2_  ~^  V  i-'i^^      i-'^-/     ^i"^ 

Show  that  <l>{z)  is  an  element  of  two  distinct  analytic  fimctions,  according  as  we 
take  I  z  I  <  I  or  I  z  I  >  j.  (  . 

10.  Show  that  a  doubly-periodic  function  /(z)  which  is  an  integral  transcen- 
dental function  is  a  constant. 

11.  Given  the  function 

Determine  the  zero  points  and  poles.     Compute  the  residue  at  each  of  the  latter. 


J. 


328  SINGLE-VALUED  FUNCTIONS  [Chap.  YU. 

12.  The  function 

f{z)  =  cos*  z  +  sin*  z 

has  the  constant  value  1  along  the  axis  of  reals.  By  what  property  does  it  follow 
that  it  must  be  equal  to  one  for  aU  finite  values  of  z?  What  other  relations  of 
trigonometric  functions  can  be  extended  in  a  sinular  manner  from  real  to  com- 
plex values  of  the  variable? 

,       -  dz  taken  along  a  circle  C  about  the  origin  as 

center.  How  large  can  the  radius  of  C  be  taken  and  have  the  value  of  the  in- 
tegral zero?    What  would  the  integral  be  if  the  radius  of  C  is  taken  §  imit  larger? 

14.  Given  a  function  having  poles  at  the  points  z  =  ai,  a2,  .  .  .  ,  ak,  and 
an  essential  singularity  at  z  =  oo.  What  kind  of  a  function  is  it?  Write  an 
infinite  series  which  will  represent  such  a  function. 

16.  By  use  of  the  theory  of  residues  evaluate  the  integral 

dx 


£ 


16.  Give  an  illustrative  example  of  a  single-valued  analytic  function  hav- 
ing (o)  no  singular  point,  (b)  no  singular  point  in  the  finite  region  and  a  pole 
at  infinity,  (c)  no  singular  point  other  than  an  essential  singular  point  at  infinity, 
(d)  a  finite  number  of  poles  in  the  finite  region  and  an  essential  singular  point 
at  infinity,  (e)  an  infinite  number  of  poles  in  finite  region  dense  at  infinity,  (/)  an 
isolated  essential  singular  point  in  the  finite  region.     Classify  each  function. 

17.  Show  that  the  infinite  product 

converges  absolutely. 

18.  Show  without  testing  the  remainder  that  the  expansion  of  tan  z  in  terms 

of  z  must  converge  for  all  values  of  z  less  in  absolute  value  than  ■^• 

19.  If  in  the  deleted  neighborhood  of  z  =  Zo,  /(z)  is  holomorphic  and  not  a 
constant,  and  if /(z)  takes  the  same  value  /3  at  a  set  of  points  dense  at  Zo,  then  z  =  Zo 
is  an  essential  singiilar  point  of /(z).     Apply  this  result  to  testing  the  nature  of 

an  -  at  z  =  0. 

z 

hint:  Consider  the  function 


7(2) -iS 

20.   Do  the  poles  of  the  function 

3  sin  z  4-  7  tan  z 


/(«)  = 


z* 


have  a  limiting  point?  What  is  the  nature  of  /(z)  at  this  point?  Show  that  /(z) 
can  be  made  to  approach  an  arbitrarily  chosen  number  as  z  approaches  this 
point.     Without  computing  the  integral 


X 


where  C  \s  the  boundary  of  a  region  S  having  z  =  oo  as  an  inner  point,  show  that 
this  int^ral  does  not  vanish. 


CHAPTER  VIII 

PROPERTIES   OF  MULTIPLE-VALUED  FUNCTIONS 

59.  Fundamental  definitions.  In  the  previous  chapters  we  have 
frequently  had  occasion  to  consider  single-valued  functions  whose 
inverse  functions  are  multiple-valued,  that  is,  are  functions  having 
in  general  two  or  more  values  for  the  same  value  of  the  independent 
variable.     The  functions 

w  =  V2,        w  =  log  z,        w  =  arc  sin  z 

are  illustrations  of  multiple-valued  functions.  In  the  present  chap- 
ter we  shall  consider  some  of  the  special  properties  of  such  functions. 
In  the  consideration  of  multiple-valued  functions  we  must  distin- 
guish between  multiple-valued  analytic  functions  and  those  multiple- 
valued  expressions  that  represent  two  or  more  single-valued  analytic 
functions.     Thus 

w  =  V22 

is  not  to  be  regarded  as  a  multiple-valued  analytic  function,  but 
rather  as  two  single-valued  functions 

w  =  z,        w  =  —z. 

These  two  functions  are  distinct  analytic  functions  rather  than  ele- 
ments of  the  same  analytic  function,  for  neither  can  be  deduced 
from  the  other  by  the  process  of  analytic  continuation. 
In  the  same  way  the  expression 

w  =  Vl  —  srn^z 

is  not  a  single  multiple-valued  analytic  function,  but  represents  the 
two  single-valued  analytic  functions 

w  =  cosz,        w  =  —  cosz. 
The  expression 

w  =  log  e* 

represents  an  infinite  number  of  distinct,  single-valued  analytic 
functions,  namely 

w  =  z,     2  +  2  iri,     2  +  4  Trt,  .  .  .  . 
329 


330  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

These  functions,  like  the  foregoing  functions,  are  distinct  because 
no  one  of  them  can  be  deduced  from  another  by  a  process  of  analytic 
continuation. 

Likewise,  the  expression  w  =  a',  where  a  =  a  -{-  ib,  is  not  a  multiple- 
valued  analytic  function  but  represents  an  infinite  number  of  single- 
valued  functions  which  can  not  be  deduced  from  each  other  by  the 
process  of  analytic  continuation.     For,  we  have 

w  =  a'  =  e^'°8°, 
which  represents  the  single-valued  analytic  functions  * 

gtloga  0t(}oga+2TX)  gilloga +4iri) 

In  mapping  by  means  of  multiple-valued  functions,  we  have 
already  seen  that  the  whole  of  one  plane  may  map  upon  a  portion  of 
the  other  plane.  For  example,  by  means  of  the  function  w  =  V2, 
the  whole  of  the  Z-plane  may  be  mapped  upon  that  half  of  the 
TT-plane  for  which  the  real  part  oi  w  =  u  -\-  iv  is  positive.  Con- 
versely, by  means  of  the  same  relation,  this  half  of  the  TF-plane  may 
be  mapped  upon  the  whole  of  the  Z-plane.  Likewise,  the  half  of  the 
TT-plane  for  which  the  real  part  of  w  is  negative  may  be  mapped 
upon  the  whole  of  the  Z-plane.  If  the  given  function  is  w  =  yz, 
the  whole  of  the  Z-plane  may  be  mapped  upon  a  sector  of  the 
TF-plane  formed  by  half-rays  drawn  from  the  origin  and  making 

angles  -, ,  respectively,  with  the  positive  axis  of  reals.     In  the 

consideration  of  multiple-valued  functions  such  as 

w  =  V2,        w  =  y/z,        w  =  log  z, 

we  have  thus  far  restricted  the  discussion  to  those  values  of  w  which 
correspond  to  values  of  z  for  which  we  have 

—IT  <  amp  z=  IT] 

that  is,  we  have  restricted  our  consideration  to  a  fundamental  region 
of  the  inverse  function.  With  this  restriction  we  have  been  able 
to  consider  these  functions  as  though  they  were  single-valued.  As 
we  shall  see,  such  an  arrangement  has  the  disadvantage  that  the 
mapping  from  at  least  one  of  the  two  planes  upon  the  other  is  not 
always  continuous. 

*  For  the  case  where  a  =  e,  we  give  to  log  e  only  the  value  1  since  e*  must  ap- 
pear as  a  special  case  of  e*.     So  also  with  other  logarithms  of  real  numbers. 


Art.  59.] 


FUNDAMENTAL  DEFINITIONS 


331 


For  example,  if  we  have  w  =  ^z,  then  by  mapping  upon  the 
TF-plane  the  amplitude  of  z  is  divided  by  three;  for,  putting 

z  =  p(cos^  +  iBind), 

w  =  p'(cos  0'  +  i  sin  6'), 
we  have  from 

*  w  =  v^, 

p'(cos  e'  +  i  sin  d')  =  p^  f  cos  5  +  i  sin  ^  j , 

^  -3 

All  points  of  the  Z-plane,  when  we  consider  only  the  principal  ampli- 
tude of  those  points,  map  into  a  region  I  (Fig.  105),  bounded  by  two 
half-rays  from  the  origin  making  an  angle  of  60°  and  —60°,  respec- 
tively, with  the  positive  half  of  the  axis  of  reals.     The  result  is  as 


Fig.  104. 


Fig.  105. 


though  the  Z-plane  were  cut  along  the  negative  half  of  the  axis  of 
reals  and  the  whole  plane  contracted  by  moving  each  bank  in  its 
own  one-half  plane  through  an  angle  of  120°  into  the  positions  of 
O'Q'  and  O'P'.  The  mapping  from  the  Z-plane  upon  the  funda- 
mental region  of  the  TT-plane  is  single-valued  but  not  necessarily 
continuous,  as  may  be  seen  most  clearly  by  considering  a  continu- 
ous curve  in  the  Z-plane  crossing  the  negative  axis  of  reals.  That 
portion  (a)  of  the  curve  above  the  X-axis  maps  into  the  curve  (a')  to 
the  right  of  O'Q',  while  the  portion  (6)  below  the  X-axis  maps  into 
(6')  to  the  right  of  O'P'.  Consequently,  what  was  a  continuous 
curve  becomes  after  mapping  a  discontinuous  curve  as  shown  in  the 
figure. 


332  MULTIPLE-VALUED   FUNCTIONS  [Chap.  VIII. 

Either  of  the  regions  II,  III  might  equally  well  have  been  selected 
as  the  particular  region  upon  which  the  Z-plane  is  mapped  by  plac- 
ing the  proper  restrictions  upon  the  variation  of  the  amplitude  of  z. 
To  any  point  a  in  the  Z-plane  distinct  from  the  origin  there  corre- 
spond then  three  points  in  the  TT-plane  when  no  restriction  is  placed 
on  the  amplitude  of  z.     We  may  denote  these  three  valueg  by  Wi(a), 

As  z  takes  all  values  in  the  Z-plane,  subject  to  the  condition  that 
—IT  <  ampz  =  TT, 

let  Wi  denote  the  corresponding  functional  values  represented  by 
the  points  of  region  I  of  the  TF-plane.  Likewise,  let  Wi,  wz  denote 
the  totality  of  values  represented  respectively  by  the  points  of  the 
regions  II,  III.  Consequently,  wi,  W2,  ws  are  single-valued  func- 
tions of  2  such  that  the  three  taken  together  give  all  of  the  corre- 
sponding values  of  w  and  z  included  in  the  functional  relation  w  =  v^z. 
We  call  wi,  W2,  ws  the  three  branches  of  the  given  function. 

Similarly,  we  may  define  a  branch  of  any  multiple-valued  analytic 
function  w  =  J{z).  An  assemblage  of  pair-values  {w,  z)  is  said  to 
define  a  branch  Wk  of  such  a  function  if  it  possesses  the  following 
properties: 

1.  The  aggregate  of  all  2-points  of  the  region  of  existence  which 
enter  into  consideration  must  fill  a  region  Sk  exactly  once. 

2.  To  each  point  of  Sk  there  corresponds  but  one  value  of  w. 

3.  The  aggregate  of  w-points  corresponding  to  points  in  Sk  repre- 
sents a  continuous  function  of  z. 

Thus,  for  the  function  w  =  "s/z  each  of  the  regions  Sk,  h  =  \,  2, 
3,  of  the  Z-plane  consists  of  the  whole  finite  portion  of  the  complex 
plane  with  the  exception  of  the  point  2  =  0,  which  is  to  be  thought 
of  as  a  boundary  point  of  Sk  for  reasons  that  will  appear  later.  Tak- 
ing the  negative  axis  of  reals  as  a  portion  of  the  boundary,  but 
nevertheless  a  part  of  Sk,  it  follows  that  the  regions  of  the  TF-plane 
corresponding  to  the  three  branches  of  the  function  are  definitely 
determined;    that  is  they  are  the  regions  I,  II,   III  in  Fig.    105. 

•       •  TV 

Had  the  amplitude  varied  between  other  limits,  say  between  —  ^ 

3  T 

and  -^r- ,  the  values  of  w  corresponding  to  values  of  2  in  >Sa;  would 

have  been  different,  and  the  branches  of  the  given  function  would 
have  been  correspondingly  changed. 


Art.  59.]  FUNDAMENTAL  DEFINITIONS  333 

The  branches  of  a  multiple-valued  analytic  function  are  single- 
valued  functions  of  the  independent  variable,  and  the  totality  of 
the  value-pairs  (w,  z),  representing  all  of  the  branches  taken  to- 
gether, is  identical  with  those  of  the  given  function.  Whenever  the 
inverse  function  z  '=  <i>{w)  is  single- valued,  then  the  domain  of  the 
functional  values  of  the  given  function  w  =  f{z)  breaks  up  into  a 
finite  or  an  infinite  number  of  non-overlapping  regions  according  as 
w  =  f(z)  has  a  finite  or  an  infinite  number  of  branches.  Moreover, 
if  in  this  case  the  given  function  has  no  natural  boundary,  each  of 
these  regions  of  the  TF-plane  may  be  taken  as  a  fundamental  region 
of  the  inverse  function. 

A  point  is  called  a  branch-point  of  the  given  analytic  function  if 
some  of  the  branches  interchange  as  the  independent  variable  de- 
scribes a  closed  path  about  it.  The  existence  of  branch-points  is  a 
characteristic  of  multiple-valued  analytic  functions  as  distinguished 
from  multiple-valued  expressions,  such  as  V 1  —  sin^  z,  representing 
two  or  more  single-valued  analytic  functions.  The  point  z  =  0  is 
a  branch-point  of  the  function  w  =  Vz.  For,  let  z  describe  a  closed 
path  about  the  origin,  beginning  at  any  point  a  whose  amplitude 
is  6,  and  let  us  consider  the  change  that  takes  place  in  the  value  of 
the  function  w  =  \^z.  The  initial  value  of  the  function  is  then 
Wi{a)  and  is  represented  by  a  point  in  the  region  I,  Fig.  105.  After 
one  revolution  of  z  about  the  origin,  the  function  does  not  return  to 
its  original  value,  but  has  changed  to  a  value  1^2(0:),  represented  by 

2x 
a  point  in  region  II,  its  amplitude  having  been  increased  by  -^• 

After  a  second  revolution  of  z  about  the  origin  the  functional  value 
has  changed  from  1^2(0:)  to  a  value  Wsia),  represented  by  a  point  in 

4  IT 

the  region  III,  and  the  amplitude  of  w  has  changed  to  6  +  -—• 

o 

After  three  revolutions  of  z  about  the  origin  the  function  again  attains 
its  initial  value  Wi{a). 

In  the  discussion  of  analytic  continuation  of  single-valued  func- 
tions, it  was  shown  that  when  the  continuation  is  taken  along  any 
path  between  two  points  Zq  and  Zi  of  a  region  *S  in  which  the  function 
is  holomorphic,  the  same  functional  value  at  Zi  is  obtained  irrespec- 
tive of  the  path  chosen,  provided  that  path  lies  wholly  within  S. 
For  multiple-valued  functions  a  different  functional  value  may  be 
obtained  at  the  terminal  point  Zi  if  the  continuation  is  taken  along 
different  paths.     This  is  always  the  case  for  some  branches  of  the 


23A  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

function  if  the  two  paths  are  so  chosen  as  to  inclose  a  branch-point; 
for,  the  two  paths  taken  together  then  form  a  closed  path  about 
that  point,  and  from  the  definition  of  a  branch-point  it  follows  that 
whenever  the  independent  variable  z  makes  but  one  circuit  of  this 
path  some  of  the  branches  of  the  function  are  interchanged  and  the 
initial  and  final  values  of  the  function  do  not  coincide.  This  prop- 
erty is,  by  the  definition  of  a  branch-point,  a  characteristic  of  all 
paths  inclosing  such  points. 

The  different  branches  of  a  function  w  =  f{z)  are  connected  with 
one  another  at  the  ly-points  corresponding  to  the  branch-points  in 
the  Z-plane.  For  example,  it  will  be  observed  that  all  three  branches 
of  the  function  w  =  V^  come  together  at  the  point  w  =  0,  which 
corresponds  to  the  branch-point  z  =  0.  As  we  shall  see  later,  how- 
ever, not  all  of  the  branches  of  an  analytic  function  need  be  con- 
nected at  any  particular  branch-point.  If  A;  4-  1  branches  coincide 
at  a  branch-point,  that  point  is  then  said  to  be  a  branch-point  of 
order  k.  In  the  illustration  discussed,  the  origin  is  therefore  a  branch- 
point of  order  two.  Since  the  branches  of  a  multiple-valued  analytic 
function  are  single-valued,  such  a  function  may  be  considered  as  an 
aggregate  of  single-valued  functions,  so  related  that  their  values  be- 
come identical  at  the  branch-points,  each  being  determined  from  the 
others  by  the  process  of  analytic  continuation. 

The  poiat  at  infinity  may  of  course  be  a  branch-point.     To  examine 

its  nature  we  may  make  use  of  the  usual  substitution  z  =  —,  and 

z 

examine  the  nature  of  the  transformed  function  at  the  origin. 

In  the  neighborhood  of  a  branch-point,  the  mapping  by  means  of 
an  analytic  function  ceases  to  be  conformal;  for,  from  the  foregoing 
discussion  it  follows  that  angles  are  not  preserved  at  such  a  point. 
In  the  case  oi  w  =  \^,  for  example,  it  was  seen  that  at  the  branch- 
point 2  =  0  angles  are  divided  by  three  in  mapping  from  the  Z-plane 
upon  the  TT-plane.  Suppose  we  have  given  a  multiple-valued  function 
w  =  f(z),  whose  inverse  function  z  =  (f>{w)  is  single- valued.  If  this 
inverse  function  fails  to  map  the  TF-plane  in  the  neighborhood  of 
Wo,  where  i^o  =  /(^o),  conformally  upon  the  Z-plane,  then  it  follows 
that  the  derived  function  <f>'(w)  must  either  be  zero  or  become  in- 
finite for  w  =  Wq.  This  fact  enables  us  to  formulate  a  convenient 
test  for  finding  the  branch-points  of  a  function  whose  inverse  func- 
tion is  single-valued.  Such  a  criterion  is  given  in  the  following 
theorem. 


^ 


Art.  59]  FUNDAMENTAL    DEFINITIONS  335 

Theorem.  Given  a  muUiple-valued  analytic  function  w  =  f{z) ,  whose 
irwerse  function  z  =  <i>{w)  is  single-valued.  If  the  derived  function  (f>'{w) 
^has  a  zero  point  of  order  k,  or  a  pole  of  order  k-\-  2,  at  Wo,  then  f(z)  has 
a  branch-point  of  order  k  at  the  corresponding  point  zq. 

Let  us  suppose  that  (f>'(w)  has  at  Wo  a  zero  point  of  order  k.  We 
may  then  write 

<l>\w)  =  (w-  WoYF,{w),  (1) 

where  A;  is  a  positive  integer  and  Fi{w)  is  holomorphic  in  the  neigh- 
borhood of  Wo  and  different  from  zero  for  w  =  Wq.  It  follows  that 
the  factor  {w  —  Wo)  enters  into  <l>iw)  —  <l>(wo)  to  a  degree  one  higher, 
that  is  to  the  degree  k  -\-  1,  thus  giving 

4>(w)  -  <t>{wo)  =  {w-  Wo^+^Fziw),  (2) 

where  Fiiw)  is  also  holomorphic  in  the  neighborhood  of  Wo  and 
different  from  zero  for  w  =  Wo.  Let  A  be  the  imncipaL  (A:  +  1)*' 
root  of  /^2(w'o)  and  let  x(j^)  be  a  function  having  the  point  i^o  as  a 
regular  point  such  that  x(wq)  is  equal  to  A.  The  function  xi'w)  can 
be  so  determined  that  for  values  of  w  in  the  neighborhood  of  Wq  we 
have 

lx{w)l''+' =  F,{w). 
From  (2)  we  now  have 

<f>(w)  -  0(Wo)  =  Kw  -  Wa)  xiw)]^'^^' 

As  a  matter  of  convenience,  we  introduce  the  auxiliary  function 

T  =  {w  -  Wo)  x{w).  (3) 

Let  us  now  suppose  the  r-plane  to  be  mapped  upon  the  TT-plane  by 
means  of  this  relation.  The  point  t  =  0  corresponds  to  the  point 
w  =  Wo,  and  the  mapping  is  conformal  in  the  neighborhood  of  r  =  0; 
because  we  have 

dwl       ^    1_1        ^  1  1        ^      1 

where  — ; — r  is  finite  and  different  from  zero,  since  x(wo)  is  the  prin- 
x(wo) 

cipal  (k  +  1)*'  root  of  F^iwo),  which  is  different  from  zero. 

If  we  now  map  the  finite  portion  of  the  Z-plane  upon  the  r-plane 

by  means  of  the  relation 

r  =    V0(u;)  -  <i>{wo)  =    Vz  -  2o,  (4) 


ur-  M^^ 


^^  ^^X-^J 


336  MULTIPLE-VALUED   FUNCTIONS  IChap.  VIIL 

the  neighborhood  of  Zo  maps  into  the  neighborhood  of  the  origin  in 
the  T-plane.  The  two  substitutions  (3)  and  (4)  are  together  equiva- 
lent to  mapping  at  once  from  the  Z-plane  to  the  TF-plane  by  means 
of  the  relation 

z  =  <i>(w), 
or  w  =  f{z). 

From  what  has  been  said  about  mapping  by  means  of  the  relation 
w  =  V  2;,  it  follows  at  once  from  (4)  that  2  =  2o  is  a  branch-point 
of  the  order  k,  which  shows  that  the  condition  stated  in  the  theorem 
is  valid  if  Wo  is  a  zero  point  of  order  k  of  <t>'{w). 

Let  us  now  consider  the  case  where  Wo  is  a  pole  of  order  /c  +  2  of 
<t>'{w).  The  corresponding  2-point  is  the  point  at  infinity.  For  this 
case  equation  (1)  takes  the  form 


<t>'{w)  = 


{W  —  WqY'^'^ 


Applying  the  same  method  as  employed  in  the  foregoing  discussion, 

it  follows  at  once  that  j{z)  has  a  branch-point  of  order  A;  at  2  =  oo  ^  >.    / -^ 

The  details  of  the  proof  are  left  as  an  exercise  for  the  student.  —  ?"*        f  / 

The  difficulties  in  representing  multiple-valued  functions  in  the  ^ 
foregoing  manner  arise  for  the  most  part  from  the  fact  that  a  con- 
tinuous curve  in  the  one  plane  does  not  always  correspond  to  a  con- 
tinuous curve  in  the  other.  Such  difficulties  can  be  easily  avoided 
by  means  of  a  simple  device  known  as  a  Riemann  surface,  consisting 
of  n  sheets  connected  with  one  another  in  a  definite  manner  depend- 
ing upon  the  character  of  the  function.  The  nature  of  such  a  surface 
can  perhaps  be  most  readily  made  clear  by  means  of  an  illustrative 
example. 

Let  us  consider  the  Riemann  surface  for  the  function 

.    w  =  Vg. 

It  has  already  been  pointed  out  (Fig.  105)  that  the  whole  of  the 
Z-plane  maps  by  means  of  this  function  into  any  one  of  the  three 
regions  I,  II,  III  of  the  PF-plane.  To  each  2-point  correspond  in 
general  three  ty-points,  one  in  each  of  the  regions  I,  II,  III.  Sup- 
pose we  think  of  the  Z-plane  as  consisting  of  three  sheets  (Fig.  106) 
connected  with  one  another  at  2  =  0  and  along  the  negative  axis  of 
reals.  As  z  describes  a  circuit  about  the  branch-point  2  =  0,  sup- 
pose it  passes  from  one  sheet  to  another  upon  crossing  the  negative 
axis  of  reals  and  that  the  variable  point  enters  the  various  sheets 


Art.  59.] 


FUNDAMENTAL  DEFINITIONS 


337 


in  the  same  order  as  the  branches  of  the  function  were  permuted  in 
the  previous  discussion  when  z  described  a  closed  path  about  the 
same  point.  Let  the  points  of  the  region  I  be  placed  into  corre- 
spondence with  those  of  the  first  sheet  of  the  Z-surface,  the  points 
of  region  II  with  those  of  the  second  sheet,  and  the  points  of  region 
III  with  those  of  the  third 
sheet.  Corresponding  to  the 
three  points  Wi(a),  1^2(0;),  Wz{pi) 
of  the  TT-plane  we  have  then 
a  point  a  in  each  of  the  three 
sheets  of  the  Z-plane,  denoted 
by  a(^),  a(2)^  q,(3)^  respectively. 
The  branch-point  0  =  0  is  not 
to  be  regarded  as  a  point  of 
the  region  of  existence  of  the 
given  function,  but  is  to  be 
considered  as  belonging  to  its 
boundary.  The  same  is  to  be 
said  of  any  branch-point  so  far 
as  the  sheets  of  the  surface  af- 
fected are  concerned.  Each  sheet  of  the  Riemann  surface,  like  the 
ordinary  complex  plane  is  to  be  regarded  as  closed  at  infinity.  By  aid 
of  the  Riemann  surface  there  is  thus  established  a  one-to-one  corre- 
spondence between  the  points  of  the  TF-plane  and  the  three-sheeted 
Z-plane.  Let  z  start  from  a  point  a(^)  in  the  first  sheet  and  describe 
a  continuous  path  about  the  branch-point  z  =  0.  By  going  once 
around  the  origin,  z  does  not  return  to  a(^)  but  changes  to  the  point 
aP^  in  the  second  sheet.  As  z  crosses  the  negative  axis  of  reals  in 
passing  from  a^^)  to  a^^),  the  corresponding  lu-point  crosses  the  line 
O'Q'  and  passes  from  the  point  W\{a)  in  the  region  I  to  the  point 
Wi{(x)  in  the  region  II.  By  a  second  revolution  about  z  =  0,  z  again 
crosses  the  negative  axis  of  reals  and  passes  into  the  third  sheet 
reaching  the  point  a(^).  At  the  same  time  the  corresponding  ty-point 
passes  from  the  region  II  to  the  region  III  reaching  the  position 
Wiipi).  After  the  third  revolution  about  the  branch-point  z  =  0, 
z  returns  to  the  first  sheet  upon  crossing  the  negative  axis  of  reals 
and  again  takes  the  value  a(^).  At  the  same  time  w  passes  across 
the  line  O'P'  and  is  again  in  region  I,  finally  assuming  its  initial 
value  i«i(a)  as  z  coincides  with  a(^).  Had  the  revolution  taken 
place  in  the  opposite  direction  about  the  branch-point,  the  order  in 


/K 


338  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

which  z  would  have  changed  sheets  would  have  been  from  the  first 
to  the  third,  from  the  third  to  the  second,  and  finally  from  the  second 
to  the  first.  The  negative  axis  of  reals  is  called  a  branch-cut,  for 
as  2  passes  over  it  in  either  direction  the  functional  values  change 

J      from  one  branch  to  another.    A 

n     cross-section  of  the  Z-plane  shows 

UI    the  connection  of  the  three  sheets. 

°'       *  If    taken   perpendicular   to    the 

negative  axis  of  reals,  the  cross-section  showing  the  intersection  of 
sheets  along  that  portion  of  the  axis  of  reals,  as  viewed  from  the 
origin,  appears  as  shown  in  Fig.  107. 

As  will  be  seen,  the  advantage  of  introducing  a  Riemann  surface 
in  place  of  the  single-sheeted  complex  plane  is  that  every  contin- 
uous curve  on  the  Z-plane  maps  by  a  multiple-valued  analytic 
function  into  a  continuous  curve  on  the  TT-plane,  and  conversely. 
This  relation  between  the  two  planes  enables  us  to  bring  to  the 
consideration  of  multiple-valued  analytic  functions  all  such  processes 
as  integration,  analytic  continuation,  etc.,  depending  on  a  continuous 
path  being  drawn  from  one  point  to  another.  In  case  w  and  z  are 
each  a  multiple-valued  function  of  the  other,  then  both  planes  are 
replaced  by  a  Riemann  surface  whose  character  is  determined  by 
the  nature  of  the  function  under  consideration.  It  is  often  con- 
venient in  such  cases  to  introduce  an  auxiliary  plane,  whereby  the 
one  Riemann  surface  may  be  mapped  upon  this  single-sheeted  com- 
plex plane  and  this  plane  in  turn  mapped  upon  the  second  Riemann 
surface.  In  the  following  articles  we  shall  consider  more  in  detail 
the  various  properties  of  Riemann  surfaces. 

60.  Riemann  surface  for  w^  —  Z  w  —  2  z  =Q.  For  any  value  of 
2  there  are  in  general  three  values  of  w;  evidently,  therefore,  there 
are  three  branches  of  the  given  function,  which  we  shall  denote  by 
Wi,  vh,  wz-  The  branch-points  can  be  at  once  determined  by  aid 
of  the  theorem  of  Art.  59.  For  2  is  a  single-valued  function  of  w, 
and  moreover  we  have 

dz      3  .   2      ^\ 

which  has  a  zero  point  of  order  one  at  w  =  -\-l,  —1,  and  a  pole  of 
order  two  sX  w  =  00 ,  Consequently,  the  given  function  must  have 
simple  branch-points  at  2  =  —  1,  -}-l,  and  a  branch-point  of  order 
two  at  2  =  00,  these  three  values  corresponding,  respectively,  to 
u>  =  +1,  —  1,  00.     That  there  can  be  no  other  branch-points  than 


Art.  60.]  BIEMANN   SURFACE  339 

these  follows  from  the  fact  that  there  are  no  other  values  of  z  for 
which  two  or  more  of  the  branches  become  equal.  The  Z-plane  then 
must  consist  of  a  three-sheeted  Riemann  surface,  the  three  sheets  all 
being  connected  at  z  =  oo ,  and  two  of  them  at  z  =  1,  and  likewise 
two  at  2  =  —1. 

The  manner  in  which  the  sheets  of  the  Riemann  surface  are  con- 
nected at  the  three  branch-points  and  a  convenient  way  for  drawing 
the  necessary  branch-cuts  can  be  determined  by  examining  the  man- 
ner in  which  the  Z-plane  may  be  mapped  upon  the  regions  of  the 
TT-plane  corresponding  to  the  three  branches  of  the  function.  The 
branches  Wi,  Wi,  W3  can  be  expressed  in  terms  of  z  by  solving  the 
given  equation 

w^-Zw-2z  =  0  (1) 

for  w  by  means  of  Cardan's  solution  of  the  cubic. 

The  general  equation  of  the  third  degree  can  be  reduced  to  the 
form 

w^  +  3Hw-\-G  =  0, 

and  Cardan's  solution  applies  equally  well  whether  H,  G,  are  real 
or  complex.*     The  three  roots  of  this  equation  are  then 

Wi  =  p  +  q,         W2  =  (jiy  -{■  00%         Wz  =  «2p  -h  cog,  (2) 

where  co  is  one  of  the  imaginary  cube  roots  of  unity  and 


subject,  however,  to  the  condition 

Vq  =  -H. 

For  the   case   under  consideration,  we  have  H  =  —1,  G  =  —2z, 

and  hence  

p  =  Vz-^  Vz2  _  1,         q  =  Vz-  V22  _  1,  (3) 

subject  to  the  condition  that  pq  =  1. 

We  shall  now  introduce  the  auxiliary  relation 

z  =  cos  3  T,  (4) 

by  means  of  which  we  obtain  from  (3) 


p  =  VcosSt-\-  Vcos^dr  -  1  =  \^cos3T-|-i  Vsin^Sr, 

q  =  Vcos  3 T  -  Vcos2  3 T  -  1  =  V cos  St  -iVsm^Sr. 
*  See  Serret,  Cours  d'alghbre  superieure,  3rd  Ed.,  Vol.  II,  p.  427. 


340  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

The  radical  Vsin^Sr  must  be  taken  with  the  same  sign,  say  the  plus 
sign,  in  both  p  and  q.  Both  p  and  q  have  three  values,  since  each  is 
the  cube  root  of  a  given  number.  Any  of  these  values  may  be  chosen 
which  satisfy  the  added  condition  pq  =  1.     We  may,  therefore,  put 

p  =  "V^cos  3  r  +  1  sin  3  T  =  cos  t  +  t  sin  t  =  e"^, 

q  =  v^cos  3  T  —  i  sin  3  T  =  cos  t  —  isinr  =  e~  *^. 

Remembering  that  co  is  an  imaginary  cube  root  of  unity,  we  may 
write 


hi  j^  2W 

0} 


u)  =  e     ,        <j}^  =  -  =  e     ^  • 


The  three  branches  Wi,  Wi,  Wz  of  the  given  function  may  now  be 
expressed  in  terms  of  t  as  follows: 

Wi  =  e*''  -\-  e~"  =  2  cos t,  (5) 

^  =  ,'  (^+t)  +  ,-K^+T^  =  2  cos  (r  +  ^),  (6) 


'<^iM) 


wz  =  e^      ^'  +  e     ^      3/=2cos(t ^1-  (7) 

The  mapping  from  the  Z-plane  upon  the  TF-plane  by  means  of 
the  given  relation  now  reduces  to  mapping  the  Z-plane  upon  the 
T-plane  by  means  of  the  inverse  of  the  function  given  in  (4)  and  then 
mapping  the  r-plane  upon  the  TT-plane  by  means  of  the  three  rela- 
tions (5),  (6),  (7).  As  we  shall  see,  the  three  branches  map  into  dis- 
tinct portions  of  the  TF-plane,  which  come  together,  however,  at  the 
points  corresponding  to  the  branch-points  z  =  —1,  +1,  oo.  We 
have  a  choice  of  the  fundamental  region  in  the  r-plane,  and  it  serves 
our  purpose  to  take  that  region  bounded  by  the  axis  of  imaginaries 
and  the  line  parallel  to  it  and  cutting  the  axis  of  reals  at  the  point 

The  whole  Z-plane  may  be  mapped  exactly  once  upon  this 


fundamental  region  of  the  r-plane.  This  fundamental  region  for  r  in 
turn  may  be  mapped  by  means  of  the  relations  (5),  (6),  (7)  into  each 
of  three  definite  regions  I,  II,  III  of  the  TT-plane. 

The  results  of  the  mapping  from  the  Z-plane  upon  the  TF-plane 
are  exhibited  m  Figs»  108  and  109.  The  details  are  left  as  an  exer- 
cise for  the  student.  By  our  choice  of  the  fundamental  region  in 
the  r-plane,  the  Z-plane  is  mapped  by  means  of  the  branch  Wi  into 


>J 


Art.  60.] 


^^<- 


Jf 


RIEMANN  SURFACE 


341 


the  region  I,  the  upper  half  of  the  Z-plane  mapping  into  the  portion 
of  this  region  above  the  axis  of  reals,  and  the  lower  half  into  the 
portion  below  that  axis.     In  a  similar  manner,  the  whole  of  the 


\.:. 


Fig.  108. 


Z-plane  maps  into  the  region  II  by  means  of  Wi  and  into  III  by 
means  of  Wz  as  indicated.  Corresponding  to  the  branch-point  z  =  1, 
we  have  w  =  —\  and  at  this  point  the  branches  wi,  Ws,  become 


W-plane 


identical.     For  z  =  —1,  we  have  w  =  1,  and  at  this  point,  as  may 
be  seen  from  Fig.  109,  the  branches  Wi,  w^,  become  identical. 

Let  us  now  think  of  the  Z-plane  as  consisting  of  three  sheets.     To 
the  first  sheet  we  associate  the  values  oi  w  in  I,  and  to  the  second 


342  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

and  third  sheets  we  then  associate  the  values  of  w  in  II  and  III, 
respectively.  As  z  traverses  a  small  closed  circuit  about  z  =  —  1 
in  the  positive  direction  starting  from  a  point  in  the  first  sheet, 
w  will  pass  from  I  to  III  and  consequently  z  passes  from  the  first 
sheet  to  the  third  sheet  of  the  three-sheeted  Riemann  surface  con- 
stituting the  Z-plane.  By  going  about  z  =  —1,  w  never  passes 
into  II,  since  only  III  and  I  come  together  at  the  corresponding  point 
w  =  1.  In  a  similar  manner,  it  will  be  seen  that  as  z  traverses  once 
a  small  closed  circuit  about  z  =  1,  beginning  at  a  point  in  the  second 
sheet,  w  passes  from  II  into  III  and  upon  continuing  a  second  time 
about  this  branch-point  w  returns  to  its  original  position  in  II.  In 
the  neighborhood  of  this  branch-point  it  is  impossible  for  z  to  pass 
from  the  second  or  third  sheet  into  the  first  sheet,  since  the  region 
I  is  not  associated  with  II,  III  at  the  corresponding  point  w  =  —1. 
From  Fig.  109,  it  is  apparent  that  all  three  branches  wi,  W2,  wz,  be- 
come identical  at  the  branch-point  z  =  ao.     The  same  result  can  be 

obtained  analytically  by  putting  z  =  —,  and  examining  the  trans- 
formed function  for  z'  =  0.  The  point  z  =  oo  is  therefore  to  be 
considered  as  belonging  to  the  boundary  of  the  region  of  existence 
on  the  Riemann  surface  and  not  as  an  inner  point  of  that  region. 

It  is  now  convenient  to  take  as  the  branch-cuts  that  portion  of  the 
axis  of  reals.  Fig.  108,  extended  from  z  =  1  indefinitely  toward  the 
right  and  from  z  =  —1  indefinitely  toward  the  left.  The  first  of 
these  segments  maps  into  the  boundary  curve  Ci  and  the  second 
into  C2  passing  through  w  =  —1,  w  =  1,  respectively,  Fig.  109. 
Along  the  axis  of  reals  between  2  =  —  1  and  z  =  1,  there  is  then  no 


-III ^     ^^ m 


Fig.  110.  Fig.  111. 

connection  between  the  various  sheets  of  the  Z-plane,  as  will  be  seen 
by  observing  the  connection  between  the  branches  of  the  function 
along  that  portion  of  the  [/-axis  into  which  this  portion  of  the  X-axis 
maps.  To  the  right  of  the  point  2=1,  the  various  sheets  of  the 
Riemann  surface  are  connected  as  shown  in  Fig.  110. 

To  the  left  of  the  point  2  =  —  1,  the  sheets  are  connected  as  shown 
in  Fig.  111. 

The  discussion  of  the  Riemann  surface  which  replaces  the  Z-plane 


Art.  61.]  RIEMANN   SURFACE  343 

for  the  given  function  is  now  complete.  Any  continuous  curve  upon 
this  surface  maps  into  a  continuous  curve  upon  the  TT-plane.  For 
example,  the  closed  curves  upon  the  Riemann  surface,  of  which  the 
ellipses  Xi,  X2,  X3  about  the  points  2=1,  —  1  as  foci  are  the  traces, 
map  into  ellipses  in  the  TF-plane.  If  the  variable  z  describes  the 
elhpse  X],  commencing  with  a  point  Zo^^^  in  the  first  sheet,  then  w 
describes  a  corresponding  path  beginning  at  i^o  lying  in  /~.  As  z 
crosses  the  positive  X-axis  the  point  continues  in  the  first  sheet  and 
w  passes  into  /+.  Upon  crossing  the  negative  X-axis,  the  point  z 
passes  from  the  first  sheet  into  the  third  sheet  and  w  passes  from  /+ 
into  III~.  When  z  has  completed  one  revolution,  it  is  still  in  the 
third  sheet  and  we  denote  its  position  by  z^^'^.  By  a  second  revolu- 
tion of  z  about  Xi,  z  passes  from  the  third  sheet  to  the  second  upon 
passing  across  the  positive  X-axis  and  remains  in  the  second  sheet  as 
it  crosses  the  negative  X-axis  ending  with  the  value  20^^).  By  a  third 
revolution  about  \\  the  point  passes  from  the  second  sheet  to  the  third 
sheet  upon  crossing  the  positive  X-axis  and  again  from  the  third  to 
the  first  sheet  upon  passing  the  negative  X-axis,  ending  with  the 
original  position  20^^^. 

61.   Riemann  surface  for  w  =  Vz  —  so+  V •     When  ration- 

y  z  —  Zi 

alized,  the  given  function  is  seen  to  be  an  algebraic  function  of  the . 
sixth  degree  in  w.  For  each  value  of  z  there  are  then  in  general  six 
distinct  values  of  w,  which  we  shall  denote  by  wi,  W2,  Ws,  wa,  w&,  Ws. 
When  the  branch-cuts  have  been  drawn  upon  the  Riemann  surface, 
the  aggregate  of  w-values  given  by  Wi,  Wi,  Ws,  Wa,  wi,,  we  in  terms  of  z 
become  definite  and  are  respectively  the  six  branches  of  the  function. 
Considered  as  functions  of  z,  we  may,  therefore,  refer  to  them  as  the 
branches  of  the  given  functions.  First  of  all  we  shall  attempt  to  dis- 
cover the  branch-points.  These  points  are  to  be  found  among  those 
values  of  z,  finite  or  infinite,  for  which  two  or  more  of  the  values  of  w 
become  identical.  Not  all  such  points  need  be  branch-points,  but 
no  other  points  can  be.  We  shall  accordingly  examine  the  points 
2  =  0,  2o,  2i,  00 .  We  can  not  make  use  of  the  theorem  of  Art.  59, 
since  2  is  not  a  single-valued  function  of  w.  We  can,  however,  de- 
termine which  of  these  points  are  branch-points  by  allowing  2  to 
describe  an  arbitrarily  small  circuit  about  each  and  observing  whether 
the  function  returns  to  its  initial  value. 
For  convenience  we  put 


344  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIIL 

The  six  values  of  w  may  be  written  in  the  form 

,  T2  ,  T2 

T3  T3 


,  T2  I    ■''2 

Wi  =  Tl  +  U  —,  Wh=    —Tl   +  W  -  , 
T3  T3 

l«3  =  Tl  +  W^  -  ,  1^6=  — Tl  +  0)2  — , 
T3  T3 


(1) 


where  w  is  an  imaginary  cube  root  of  unity.  Let  z  describe  a  circuit 
about  z  =  0,  taken  sufficiently  small  to  exclude  both  Zo  and  Zi.  Since 
gi  =  T2,  it  follows  that  by  one  revolution  of  the  circuit  by  z,  t2  rotates 

4  IT 

thrgugb-  an  angle^  -^ ,  which  is  equivalent  to  multiplying  t2  by  co^. 


Hence,  since  ti  and  r*  return  to  their  original  values  after  each  revo- 
lution, Wi  is  changed  mto  Ws,  Wz  into  w^,  Wi  into  Wi,  Wi  into  we,  we  into 
Wi,  Wi  into  «;4.  By  a  second  revolution  of  the  circuit  a  similar  change 
takes  place,  and  upon  a  completion  of  the  third  revolution,  the  orig- 
inal values  are  restored.     The  results  may  be  exhibited  as  follows: 

Before  z  changes,  we  have  wi,  Wz,  Ws;  Wi,  w^,  we; 

after  one  revolution,  we  have  W3,  Wi,  w^;  w^,  W4,  w^; 

after  two  revolutions,  we  have  W2,  Ws,  Wi;  W5,  We,  w^; 

after  three  revolutions,  we  have  Wi,  W2,  ws]  Wi,  ws,  We. 

The  point  2  =  0  is  therefore  a  branch-point.  It  will  be  seen  that 
the  six  branches  form  two  sets,  Wi,  w^,  W3  and  wt,  w^,  W6,  each  of  which 
is  cyclically  permuted  by  successive  revolutions  of  z  about  2  =  0. 
By  successive  revolutions  about  this  point,  the  branches  constituting 
the  first  set  do  not  pass  into  those  of  the  second. 

In  a  similar  manner  we  can  test  the  point  z  =  Zi.  By  each  revolu- 
tion of  z  about  this  point  ti,  n  remain  unchanged,  while  ts  is  multiplied 
by  o).     Remembering  that  the  factor  ts  appears  in  the  denominator 

and  that  -  =  ul^,  -5  =  to,  -5  =  1,  we  have,  as  the  result  of  the  succes- 

sive  revolutions  about  21,  the  same  cyclical  interchange  of  the  various 
values  of  w  as  about  the  origin.  Consequently,  the  point  z  =  Zi  is 
hkewise  a  branch-point. 

As  2  describes  an  arbitrarily  small  circuit  about  Zo,  the  values  of 
T2,  t3  remain  unchanged  but  ti  changes  sign  by  each  revolution. 
The  results  are  as  follows: 

Before  2  changes,  we  have  Wi,  w^,  Wz,  Wi,  w^,  we; 

after  one  revolution,  we  have  1^4,  Ws,,  w^,  Wi,  W2,  Wz] 

after  two  revolutions,  we  have  Wi,  Wt,  Wz,  W4,  Wh,  wn. 


Art.  61.] 


RIEMANN  SURFACE 


345 


Consequently,  w  =  Zo  is  also  a  branch-point.  All  of  the  branches 
are  affected,  but  they  form  three  sets,  each  set  including  two  branches, 
namely : 

Wi  and  Wi,        w^.  and  w^,         wa  and  w^. 

To  examine  the  point  2  =  oo ,  we  put  z  =  —  and  have 


w 


-sir- 


20  + 


vr^ 


2o2 


Vz 


<^ 


Z\Z 


</z 


Putting 

X2, 

Vl  -  z'zo  =  Xi,         Vz'  = 

VI  -  z,z'  =  X3,      V 

we  have  for  the  six  values  of 

w' 

^  ,  _  Xi        1 
X2      X3X4 

w'  -■     ^'  +    ^ 
X2      X3X4 

/      Xi   ,        1 

A2             A3A4 

Xi  ,        1 

A2            A3A4 

/   _  Xl     ,         2      ^ 

X2                 X3X4 

/    _            Xl               2       1 

X2                  X3X4 

<^z'  =  X4, 


As  z'  describes  an  arbitrarily  small  circuit  about  the  origin,  Xi,  X3 
are  not  changed,  X2  changes  sign  and  X4  is  multiplied  by  the  factor  co, 
where  co,  as  before,  is  an  imaginary  cube  root  of  unity.  The  results 
of  successive  revolutions  about  this  circuit  are  shown  in  the  following 
table 


Before  z'  changes,  we  have 

i^/ 

after  one  revolution. 

W 

after  two  revolutions, 

Wi' 

after  three  revolutions, 

w 

after  four  revolutions. 

Ws, 

after  five  revolutions, 

Wff, 

after  six  revolutions, 

W\ 

',  W2, 

W3, 

W4 

,Wi' 

We'; 

',W 

W5' 

Wi 

,Wi' 

Wi'; 

,w, 

Wi, 

Wi 

,we', 

Wi'; 

',  Wb', 

We, 

Wi 

,W2' 

Wi'; 

,Wi', 

W2, 

We 

,  w, 

w&'; 

,W6, 

Wi, 

W2 

yWz', 

Wi'; 

,W2, 

W3, 

Wi 

,W5, 

We'. 

It  follows  that  z'  =  0  and  therefore  z  =  00  is  a  branch-point  where 
all  of  the  sheets  of  the  Riemann  surface  are  associated  with  one 
another. 

The  Riemann  surface  constituting  the  Z-plane  has  then  the  branch- 
points z  =  0,  Zo,  01,  00.     No  two  of  the  finite  branch-points  can  be 


346  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

connected  with  a  branch-cut.     At  first  thought,  it  may  seem  possible 
to  connect  the  two  finite  branch-points  z  =  0  and  z  =  Zy  with  a  branch- 

cut  since  the  same  sheets  are  con- 
IMl I 1 ??1 nected  in  the  same  way  at  these 

O  231      1  I         213      Zi  .  u  •  + 

^  T  two  points.     This,  however,  is  not 

possible;  for,  by  successive  revolu- 
tions about  each  of  these  points  the  sheets  would  be  connected  along 
such  a  branch-cut  as  shown  in  Fig.  112. 

Then,  by  crossing  the  branch-cut,  to  the  right  of  z  =  0  in  the 
direction  of  arrow  (a)  we  would  pass,  for  example,  from  the  first 
sheet  to  the  second  sheet,  while  in  passing  across  the  branch-cut 
to  the  left  oi  z  =  Zi  in  the  same  direction  we  would  pass  from  the  first 
sheet  to  the  third  sheet.  Such  connection  of  the  sheets  along  a 
branch-cut  is  impossible  since  the  connection  must  be  between  the 
same  sheets  along  its  entire  length.  It  is  at  once  apparent  that  in 
order  to  connect  any  two  finite  branch-points  with  a  branch-cut, 
the  cychcal  interchange  of  branches  must  involve  the  same  sheets  but 
in  reverse  order.  We  can,  however,  always  draw  branch-cuts  from 
any  branch-point  to  the  point  2  =  go  ;  that  is,  the  branch-cuts  may 
be  taken  as  lines  extending  out  indefinitely  from  the  branch-points. 
Drawing  these  lines  in  any  convenient  manner,  the  six  branches  of 
the  given  function  are  fully  determined  by  the  six  definite  aggregates 
of  value-pairs  (to,  z),  such  that  to  the  values  of  z  associated  with  each 
sheet  of  the  Riemann  surface,  there  is  a  definite  branch  of  the  func- 
tion. In  this  case,  however,  there  are  no  corresponding  fundamental 
regions  in  the  PT-plane,  since  the  inverse  function  is  not  single- valued. 
The  general  discussion  of  the  Riemann  surface  required  for  the  Z- 
plane  is  now  complete. 

62.  Riemann  surface  for  w  =  log  z.  The  logarithm  is  a  func- 
tion having  an  infinite  number  of  branches.  As  we  have  seen  the 
logarithm  of  z  may  be  written  in  the  form 

w;  =  log  2  =  log  p  +  iB,  (1) 

where 

2  =  p(cos0  +  isin^). 

By  means  of  the  relation  (1)  the  whole  of  the  Z-plane  maps  into  any 
one  of  the  strips 

{2k-l)Tr  <v^  {2k  +  l)ir 

of  the  TF-plane  parallel  to  the  axis  of  reals.     Conversely,  any  one  of 
this  infinite  number  of  strips  maps  into  the  whole  of  the  Z-plane. 


Art.  63.]  BRANCH-POINTS,   BRANCH-CUTS  347 

We  may  now  replace  the  Z-plane  by  a  Riemann  surface  having  an 
infinite  number  of  sheets.  The  point  z  =  0  is  a  branch-point  of  the 
surface;  for,  as  we  see  from  (1),  every  revolution  of  2  in  a  positive 
direction  about  a  circle  having  the  origin  as  a  center  leaves  p  un- 
changed but  increases  ^  by  2  tt,  thus  changing  the  value  of  w.  Thia 
change  continues  indefinitely  by  successive  revolutions.  We  may 
take  the  negative  half  of  the  axis  of 

reals  as  the  branch-cut,  so  that  every  ^ 

time  the  variable  point  crosses  this  ^--^^ 

part  of  the  axis  it  passes  from  one  ^"^"^ 

sheet  to  the  next  succeeding  or  next  -^^ 

preceding  sheet,  according  as  5  is  Fi     113 

increasing  or  decreasing.     A  cross- 
section  of  the  surface  perpendicular  to  the  negative  half  of  the  axis 
of  reals,  as  seen  from  the  origin,  is  then  of  the  form  shown  in  Fig. 
113.     In  the  same  way  the  point  z  =  cc  may  be  shown  to  be  also  a 
branch-point  of  infinitely  high  order. 

The  inverse  function  of  to  =  log  z  is,  as  we  know,  z  =  e^.  The 
trigonometric  functions  were  defined  in  terms  of  the  exponential 
function,  so  that  the  general  character  of  the  Riemann  surfaces  con- 
nected with  the  inverse  trigonometric  functions  may  be  easily  de- 
duced by  aid  of  the  logarithmic  function.*  In  the  discussions  to 
follow  we  shall  have  occasion  to  consider  for  the  most  part  only 
Riemann  surfaces  having  a  finite  number  of  sheets. 

63.  Branch-points,  branch-cuts.  Having  discussed  some  typi- 
cal illustrations  of  Riemann  surfaces,  we  shall  now  consider  some  of 
the  more  important  properties  of  branch-points  and  branch-cuts  and 
their  relation  to  the  Riemann  surfaces  needed  in  the  representation 
of  multiple-valued  functions. 

The  branch-points  of  a  function  are  always  to  be  found  among 
those  points  corresponding  to  the  values  of  the  independent  variable 
for  which  two  or  more  of  the  values  of  the  function  become  equal. 
This  common  value  of  the  various  branches  of  the  function  may  be 
finite  or  infinite.  As  we  have  seen,  not  all  of  the  various  sheets  of  a 
Riemann  surface  need  be  connected  at  any  particular  point;  for, 
they  may  be  associated  in  distinct  sets  at  a  branch-point  as  was  the 
case  in  the  points  z  =  Zo,  z  =  0,  z  =  Zi  for  the  function  discussed  in 
Art.  61.     It  is  essential,  however,  that  all  of  the  sheets  be  so  con- 

*  Cf.  Fouet,  Lec^ons  dementaires  sur  la  theorie  des  fonctions  analytiques,  2d 
Ed.,  Tome  II,  p.  128  et  seq. 


348  MULTIPLE-VALUED   FUNCTIONS  [Chap.  VIII. 

nected  at  the  various  branch-points  that  the  entire  surface  forms  a 
connected  whole  and  it  is  possible  to  proceed  along  a  continuous  path 
from  any  point  to  any  other  point  upon  the  surface.  As  already 
pointed  out,  the  branch-points  themselves,  while  points  of  the  Rie- 
mann  surface,  are  not  to  be  regarded  as  points  of  the  region  of  exist- 
ence of  the  given  function,  but  as  boundary  points  of  that  region. 
The  region  of  existence  is  then,  as  with  single-valued  analytic  func- 
tions, to  be  considered  an  open  region,  which  in  the  case  of  multiple- 
valued  functions,  however,  extends  to  the  several  sheets  composing 
the  Riemann  surface.  It  is  convenient  to  speak  of  the  branch-points 
as  points  of  the  function  in  the  sense  that  their  character  aids  in 
describing  the  character  of  the  function. 

Whenever  the  inverse  function  is  single-valued,  we  have  in  the 
theorem  of  Art.  59  a  convenient  test  for  locating  the  branch-points 
and  determining  their  order.  This  test  is  not,  however,  valid  when 
we  have  as  in  Art.  61  a  function  whose  inverse  is  multiple-valued. 
In  that  case  it  is  necessary  to  test  each  point  where  some  of  the 
values  of  the  function  become  identical  by  permitting  the  indepen- 
dent variable  to  traverse  an  arbitrarily  small  closed  circuit  about  the 
point  and  observe  whether  the  corresponding  values  of  the  function 
return  to  the  initial  value. 

In  the  illustrative  examples  discussed,  it  was  observed  that  the 
branches  of  a  function  associated  with  each  other  at  a  branch-point 
were  cyclically  interchanged  by  successive  revolutions  about  the 
point.  As  a  matter  of  fact,  it  is  always  true  that  all  of  the  branches 
affected  at  a  branch-point  can  be  so  arranged  that  by  successive 
circuits  of  the  independent  variable  about  the  point  these  branches 
are  cyclically  interchanged.  The  cycle  may  include  all  or  only  a 
portion  of  the  branches  affected.  In  the  latter  case  there  may  be 
two  or  more  sets  each  of  which  undergoes  a  cyclical  interchange  as 
the  independent  variable  traverses  a  circuit  about  the  branch-point 
a  suitable  number  of  times.  To  show  that  the  branches  of  the  func- 
tion are  interchanged  as  stated,  suppose  we  let  the  branches  affected 
be  wi,  u>2,  wz,  .  .  .  .  Then  as  z  describes  a  small  circuit  about  the 
branch-point,  wi  must  change  into  some  other  branch,  say  t02.  By 
a  second  revolution  about  the  point,  it?2  can  not  remain  unchanged; 
for,  in  that  case,  tracing  the  circuit  in  a  reverse  order  would  not 
restore  the  initial  value  Wi,  as  it  should  do.  It  is  possible  that  this 
second  revolution  changes  the  branch  'w2  back  into  Wi,  in  which  case 
Wi,  vh  constitute  a  complete  cycle.     If  this  is  not  the  case  then  Wi 


Abt.  63.]  BRANCH-POINTS,   BRANCH-CUTS  349 

must  change  into  some  one  of  the  remaining  branches,  say  W3.  As 
before,  when  2  describes  the  circuit  about  the  branch-point  a  third 
time  W3  can  not  remain  unchanged  and  can  either  change  back  into 
Wi  or  into  some  one  of  the  remaining  branches,  say  Wi.  In  the  first 
case,  the  three  branches  wi,  W2,  Wz  constitute  by  themselves  a  set 
which  cyclically  interchange.  In  the  other  case,  Wi  changes  in  a 
similar  way  into  w-o,  say,  by  another  revolution  about  the  branch- 
point. This  method  of  interchange  may  involve  all  of  the  branches 
in  one  set,  or  in  several  such  sets.  There  may  be  other  sheets  that 
are  not  affected  at  this  particular  branch-point,  the  connection  with 
one  or  more  of  the  remaining  sheets  being  made  at  another  branch- 
point. 

The  same  sheets  may  be  affected  at  two  branch-points.  If  the 
order  of  the  cyclic  permutation  of  the  sheets  in  the  one  case  is  the 
reverse  of  what  it  is  in  the  other,  then  it  is  always  possible  to  connect 
the  two  points  with  a  branch-cut;  for,  by  crossing  this  cut  at  any 
point,  the  variable  passes  from  one  sheet  into  another  particular 
sheet  of  the  cycle.  Instead  of  connecting  the  branch-points  with 
each  other  it  is  always  possible  to  connect  the  various  branch-points 
with  the  point  at  infinity  with  branch-cuts.  Thus  far  in  the  dis- 
cussion the  various  branch-cuts  have  been  taken  as  straight  lines. 
This  restriction  is  unnecessary,  however,  as  the  choice  of  a  particular 
curve  for  the  branch-cut  is  a  purely  arbitrary  convention.  A 
branch-cut  is,  however,  always  to  be  taken  so  that  it  has  no  double- 
points,  that  is,  the  cut  must  not  intersect  itself.  While^  the  partic- 
ular aggregate  of  value-pairs  {w,  z)  constituting  the  various  branches 
are  changed  by  varying  the  branch-cut,  the  connection  of  the 
branches  with  each  other  at  the  branch-points  remains  unchanged. 
The  following  illustration  will  make  this  statement  clear. 

Ex.  1.  Discuss  the  branch-cuts  of  the  Riemann  surface  required  for  the  func- 
tion w  =    V  ^"i — 

y  %-\-z 

The  Z-plane  is  a  double-sheeted  Riemann  surface  and  the  points  z  =  i  and 
z  —  —  i  are  simple  branch-points.  The  corresponding  functional  values  are  re- 
spectively u;  =  0  and  tw  =  oo.  The  result  of  mapping  the  Z-plane  upon  the 
TT-plane  is  exhibited  in  Figs.  114,  115.  Any  curve  in  the  Z-plane  joining  the 
points  i  and  —i  may  be  taken  as  a  branch-cut  and  will  map  into  a  curve  in  the 
TF-plane  joining  the  points  ?i;  =  0  and  1^  =  00.  For  example,  if  we  select  that 
portion  of  the  axis  of  imaginaries  joining  the  two  branch-points,  then  the  cor- 
responding curve  in  the  W-plane  is  the  positive  half  of  the  axis  of  reals.  To 
every  point  in  the  Z-plane,  however,  correspond  two  points  in  the  PT-plane  and 


'J  t^ 


UT- 


.J^^' 


350 


MULTIPLE-VALUED  FUNCTIONS 


[Chap.  VIII. 


this  same  portion  of  the  axis  of  imaginaries  also  maps  into  the  negative  half  of 
the  axis  of  reals  in  the  TF-plane.  Consequently,  in  this  case  the  two  branches 
Wi,  W2  of  the  function  corresp)onding  to  the  values  of  z  in  the  first  and  second 
sheets  of  the  Z-plane  are  represented  by  points  in  the  upper  and  the  lower  half 
of  the  TF'-plane  respectively.  If  we  take  any  circle  through  i  and  —i  and  select 
that  portion  (2)  lying  to  the  right  of  the  Une  x  =  0  as  the  branch-cut,  then 


Z-plane 


W-plane 


Fia.  114. 


Fig.  115. 


the  corresponding  curve  in  the  TT-plane  is  a  half-ray  (2')  proceeding  from  the 
origin.  The  same  curve  (2)  also  maps  into  the  half-ray  making  an  angle  of  180° 
with  (2').  The  two  half-rays  taken  together  divide  the  TT-plane  into  two 
regions  and  the  points  in  these  two  regions  represent  the  values  of  w  corre- 
sponding to  the  two  branches  of  the  given  fimction.  If  instead  of  a  circle  through 
t  and  —i  any  ordinary  curve  had  been  taken  as  a  branch-cut,  it  would  map 
into  a  curve  joining  the  points  w  =  0  and  w  =  <x>  and  again  into  a  congruent 
curve,  which  may  be  obtained  by  rotating  the  first  curve  through  an  angle  of 
180°.  These  two  curves  taken  together  constitute  a  continuous  curve  dividing 
the  TF-plane  into  two  regions  whose  points  give  the  values  of  the  two  branches 
Wi,  Wo  of  the  function  corresponding  to  the  two  sheets  of  the  Riemann  surface 
constituting  the  Z-plane.  Again  we  may  select  as  the  branch-cuts  of  the  Rie- 
mann surface  any  curves  joining  the  two  branch-points  i  and  —i  with  the  point 
2  =  00,  for  example  those  portions  of  the  axis  of  imaginaries  exterior  to  the  circle 
of  unit  radius  about  the  origin.  This  selection  likewise  leads  to  a  division  of  the 
TT-plane  into  two  regions  and  corresponding  branches  of  the  function. 

From  this  discussion,  it  will  be  seen  that  the  branch-cuts  can  be 
selected  in  a  variety  of  ways  and  may  or  may  not  be  straight  lines. 
By  the  selection  of  the  branch-cuts  particular  values  of  the  function 
constituting  the  various  branches  are  determined.  The  number  and 
the  association  of  such  branches  are  determined  by  the  character  of 
the  function  itself.  While  the  selection  of  the  branch-cut  is  arbi- 
trary, there  is  often  an  advantage  in  selecting  it  in  a  particular  manner. 
For  example,  in  the  discussion  of  the  Riemann  surface  for  w  =  v^ 


\Ar  '-  i(l.-\-C)(2-() 


^  - 


fj±. 


-i^o- 


Art.  63.]  BRANCH-POINTS,   BRANCH-CUTS  351 

we  chose  the  negative  half  of  the  axis  of  reals  as  the  branch-cut  in 
order  that  one  branch  of  the  function  should  correspond  to  the  prin- 
cipal value  of  the  amplitude  of  z.  Again  in  the  logarithmic  function 
the  negative  half  axis  was  chosen  for  the  same  reason.  In  discussing 
the  inverse  of  a  periodic  function,  it  is  hkewise  an  advantage  to  select 
the  branch-cuts  so  that  the  previously  determined  fundamental 
regions  shall  correspond  to  single  sheets  of  the  Riemann  surface 
rather  than  conversely. 

The  following  theorems  concerning  branch-points  and  branch-cuts 
give  additional  information  that  will  be  useful  in  the  discussion  of 
special  Riemann  surfaces. 

Theorem  I.     A  free  end  of  a  branch-cut  is  a  branch-point. 

Let  a  be  a  free  end  of  a  branch-cut  C.  Suppose  that  as  z  crosses 
this  branch-cut  it  passes  from  the  ki^^  sheet  into  the  k2^''.  Then  as  z 
makes  a  complete  circuit  about  a  starting  from  an  initial  position 
Zo'-^'^  in  the  fci'*  sheet  it  does  not  return  at  the  end  of  the  first  revolution 
to  that  initial  position,  but  it  ends  in  a  point  Zo^*'^  in  the  ^2"*  sheet. 
Hence,  from  the  definition  of  a  branch-point,  a  is  such  a  point. 


Fig.  116.  Fig.  117. 

It  is  to  be  noted  that  in  case  a  branch-cut  ends  in  a  point  on  the 
boundary  of  the  region  of  existence  it  is  not  necessarily  a  branch- 
point. 

Theorem  II.  If  but  one  branch-cut  passes  through  a  branch-point, 
the  connection  of  the  sheets  on  the  two  sides  of  the  branch-point  is  not 
the  same. 

Let  a  be  a  branch-point  through  which  but  one  branch-cut  passes. 
The  connection  between  the  sheets  can  not  be  the  same  on  the  one 
side  of  a  as  on  the  other;  for,  in  that  case  as  the  variable  z  describes 
a  circuit  about  a  it  returns  after  each  revolution  to  its  initial  posi- 
tion. For  suppose  the  /ci"'  and  ^2^''  sheets  are  connected  along  the 
given  branch-cut  C,  as  indicated  in  Fig.  117.     If  2  has  the  initial 


352  MULTIPLE-VALUED   FUNCTIONS  [Chap.  VIII. 

value  Zo^*'^  and  describes  a  circuit  about  a,  say  in  the  positive  direc- 
tion, then  as  z  crosses  C  at  a  it  passes  from  the  k\^^  sheet  to  the  A^'*. 
Upon  crossing  the  branch-cut  again  at  6,  z  passes  from  the  k^^  sheet 
back  to  the  fci'*  sheet  returning  to  its  initial  value  2o^*'^- 

Theorem  III.  7/  two  branch-cuts,  having  different  sequences  of 
interchange  of  sheets  associated  with  them,  meet  in  a  point,  that  point  is 
either  a  branch-point  or  is  an  extremity  of  at  least  one  other  branch-cut. 

Let  a  be  the  common  point  of  the  two  branch-cuts,  I,  II,  Fig,  118, 
Along  the  branch-cut  I,  let  the  ki""  sheet  be  associated  with  the  k2"' 

and  along  II,  since  the  sequence  of  inter- 
change of  sheets  can  not  be  the  same, 
suppose  the  ki""  to  be  connected  with  a 
third  sheet  ks"'.  If  the  variable  starts 
from  an  initial  point  2o^*'^  in  the  ki'''  sheet, 
then  by  passing  around  a  in  a  positive 
direction  it  passes  into  the  /ca'^  sheet, 
upon  crossing  I.  Upon  crossing  II  it 
passes  from  the  A^a"*  sheet  to  the  /^a"'  sheet  and,  if  it  crosses  no  fur- 
ther branch-cut,  then  instead  of  returning  to  the  initial  position  Zo^*'^ 
at  the  close  of  the  first  circuit  it  ends  in  a  point  Zo^*'^  in  the  ks^''  sheet 
and  a  is  a  branch-point.  Hence  either  a  is  a  branch-point  or  there 
must  be  at  least  one  more  branch-cut  like  III  ending  in  a  by  which 
the  ks""  sheet  is  connected  with  the  A;i"'  sheet. 

Theorem  IV.  If  a  change  of  sequence  in  the  branches  of  a  function 
occurs  at  any  point  of  a  branch-cut,  then  that  point  is  a  branch-point  or 
it  lies  also  on  some  other  branch-cut. 

Since  by  hypothesis  a  change  of  sequence  occurs  at  some  point, 
say  a,  then  as  z  describes  a  circuit  about  a  it  does  not  return  to  the 
sheet  from  which  it  started  but  passes  into  another  sheet  of  the  sur- 
face. Hence,  the  point  a  must  either  be  a  branch-point  or  another 
branch-cut  must  terminate  at  a. 

Theorem  V.  A  branch-cut  passing  through  but  one  branch-point 
can  not  be  a  closed  curve. 

If  a  branch-cut  having  but  one  branch-point  is  a  closed  curve,  then 
by  encircling  that  point  along  an  arbitrarily  small  circuit,  the  variable 
returns  to  the  same  sheet;  for,  the  connection  between  the  sheets  must 
be  the  same  on  both  sides  of  the  branch-point,  since  the  portions  of 


Art.  63.] 


BRANCH-POINTS,   BRANCH-CUTS 


353 


the  cut  on  the  two  sides  of  the  point  belong  to  the  same  branch-cut. 
It  is,  however,  impossible  for  the  variable  to  encircle  a  branch-point 
and  not  change  sheets.  Hence  under  the  conditions  set  forth  in  the 
theorem,  the  branch-cut  can  not  be  a  closed  curve. 

In  general  we  have  considered  only  paths  which  encircle  a  single 
branch-point.  In  Art.  60  we  considered  certain  paths  encircling 
two  branch-points.  We  shall  now  consider  the  general  effect  of  a 
path  encircling  two  or  more  branch-points.  We  have  the  following 
theorem. 

Theorem  VI.  The  effect  of  describing  a  closed  circuit  about  several 
branch-points  is  the  same  as  though  the  variable  point  had  described  a 
dosed  path  about  each  of  the  branch-points  in  succession* 

We  shall  prove  the  theorem  for  the  case  of  a  circuit  about  three 
branch-points.  The  same  argument  holds  for  any  finite  number  of 
such  points.  It  is  at  once  evident  that  a  path  can  be  deformed  in 
any  manner  without  affecting  the  result,  provided  in  such  a  defor- 
mation a  branch-point  is  not  encountered.  For  by  such  a  deforma- 
tion no  additional  branch-cuts  need  be  crossed  an  odd  number  of 
times.  If  crossed  an  even 
number  of  times,  the  final 
position  of  the  variable 
point  is  in  the  same  sheet 
as  the  initial  point. 

Consequently  the  closed 
path   C    about    the    three 
points  Zo,  Zi,  Z2,  can  be  de- 
formed   without    affecting  ^^ 
the    final    result    into    the                              Fig.  119. 
succession  of  paths  Xo,  Xi, 

X2  about  the  three  points  Za,  Zi,  22,  respectively.  Each  of  these 
paths  begins  and  ends  at  Zs,  as  shown  in  Fig.  119,  and  consists  of 
a  small  circle  about  the  branch-point  and  a  path  connecting  that 
circle  with  23.  This  connecting  path,  however,  is  traversed  twice, 
once  in  each  direction,  so  that  any  branch-cut  crossed  in  going  in 
one  direction  will  be  crossed  again  in  an  opposite  direction  when  the 
path  is  traversed  in  the  opposite  direction.     The  effect  of  this  por- 

*  The  group  of  permutations  which  the  function  values  undergo  as  the  inde- 
pendent variable  describes  a  closed  path  is  often  called  a  monodromic  group.  Cf. 
Encyklopddie  der  Math.  Wiss.,  Bd.  II2,  p.  121. 


354  MULTIPLE-VALUED   FUNCTIONS  [Chap.  VIII. 

tion  of  the  path  crossing  a  branch-cut  can  therefore  be  neglected. 
The  total  effect  of  traversing  the  closed  path  C  is  then  the  same  as 
traversing  in  succession  the  closed  circuits  about  the  separate  branch- 
points, as  stated  in  the  theorem. 

The  foregoing  theorem  also  gives  us  a  convenient  way  for  testing 
the  point  2  =  oo  for  a  branch-point.  Consider  a  closed  path  inclos- 
ing all  of  the  finite  branch-points.  If  this  path  be  traversed  in  a 
counter-clockwise  direction  the  result  is  easily  obtained  by  the 
theorem.  However,  since  no  finite  branch-point  lies  exterior  to  this 
path,  it  follows  that  traversing  the  path  in  a  clockwise  direction  is 
equivalent  to  encircling  the  point  at  infinity.  Traversing  the  path 
in  a  clockwise  direction  gives  the  same  interchange  of  sheets  but  in 
opposite  order  as  traversing  it  in  the  opposite  direction.  Therefore, 
if  the  sheets  are  interchanged  by  traversing  such  a  path  in  either 
direction  the  point  2  =  00  is  a  branch-point. 

Ex.2.  Given  the  function  w  =  V{z  —  zq){z —  z{).  Determine  whether  the 
point  z  =  00  is  a  branch-point. 

The  given  function  has  a  branch-point  of  the  second  order  at  2  =  Zo  and  at 
z  =  Zu  At  2o  and  Zi  the  branches  interchange  by  successive  clockwise  revolutions 
about  the  point  as  follows:  ~~ 

Before  z  changes,  Wi,  wi,  w^]  ^^---.^ 

after  one  revolution,  Wi,  wz,  Wi] 

after  two  revolutions,  103,  Wi,  xjih', 

after  three  revolutions,  wi,  Wi,  wi. 

As  z  describes  clockwise  a  closed  circuit  inclosing  both  20,  2i,  we  have: 
Before  z  changes,  u;i,  Wi,  wz; 

after  one  revolution,  wz,  Wi,  vh) 
after  two  revolutions,  Wi,  wz,  Wi] 
after  three  revolutions,     W\,  W2,  Wt. 

The  point  z  =  00  is  then  a  branch-point  of  order  two. 

It  is  to  be  observed  that  had  the  cycUc  arrangement  of  the  sheets  at  Zo,  Z\  been 
such  that  these  points  might  have  been  connected  by  a  branch-cut,  then  there 
would  have  been  no  interchange  of  branches  as  2  described  a  closed  circuit  about 
the  two  branch-points  and  consequently  the  point  z  =  00  would  not  have  been 
a  branch-point.  In  order  that  the  two  points  Zo  and  zi  might  have  been  so  con- 
nected the  cyclic  arrangement  of  the  branches  at  this  point  would  necessarily 
have  involved  the  same  sheets  taken  in  opposite  order.  Such  would  be  the  case, 
for  example,  with  the  fimction  

»*/z  -  Zo 
w  =  \ •  • 

T   Z  —  Zi 

64.  Stereographic  projection  of  a  Riemann  surface.  As  with 
single-valued  functions,  stereographic  projection  upon  a  sphere  may 


Abt.  65.]  PROPERTIES  OF  RIEMANN  SURFACES  355 

often  be  employed  with  advantage  in  the  discussion  of  multiple- 
valued  functions.  The  multiple-sheeted  Riemann  surface  projects 
into  a  multiple-sheeted  Riemann  sphere  whose  sheets  are  associated 
at  the  branch-points.  The  branch-cuts  in  the  plane  go  over  into 
curves  upon  the  sphere  along  which  the  variable  point  passes  from 
one  sheet  to  another. 

In  the  case  of  the  inverse  of  the  exponential  and  trigonometric 
functions,  namely  the  logarithmic  and  inverse  trigonometric  func- 
tions, the  projection  of  the  TF-plane  upon  the  sphere  is  also  of  inter- 
est. The  infinite  number  of  strips  congruent  with  the  fundamental 
strip  map  into  similar  regions  having  a  common  point  and  bounded 
by  curves  having  a  common  tangent  at  the  north  pole.  This  result 
exhibits  the  fact  that  the  exponential  function  e"*,  in  terms  of  which 
the  other  functions  are  defined,  is  a  function  which  takes  in  the  neigh- 
borhood of  the  essential  singular  point  w  =  oo  every  complex  value 
except  zero  and  infinity.  The  branch-points  may  be  regarded  as 
boundary  points  of  the  region  of  existence  on  the  sphere  just  as  they 
are  regarded  upon  the  Riemann  surface. 

65.  General  properties  of  Riemann  surfaces.  Thus  far  we  have 
considered  only  special  cases  of  Riemann  surfaces.  In  general  to  con- 
struct such  a  surface  for  a  given  function,  we  may  proceed  as  follows. 

1.  Determine  the  number  of  branches.  This  number  is  equal  to 
the  largest  number  of  distinct  values  which  the  function  has  for  each 
value  of  the  independent  variable.  In  case  the  function  is  algebraic, 
the  number  of  branches  is  equal  to  the  degree  of  the  algebraic  equa- 
tion defining  the  function,  when  that  equation  is  freed  from  radicals. 

2.  Locate  the  branch-points.  If  the  inverse  of  the  given  function 
is  single-valued,  the  branch-points  may  be  found  by  the  theorem  of 
Art.  59.  In  any  case,  they  are  to  be  found  among  the  points  of  the 
complex  plane  representing  values  of  the  independent  variable  for 
which  two  or  more  values  of  the  function  are  equal,  and  they  may  be 
either  all  at  finite  points  or  one  of  them  may  be  the  point  at  infinity. 
From  among  these  points  those  that  are  branch-points  may  be  found 
by  permitting  the  independent  variable  to  describe  a  closed  path 
about  each  point  and  observing  which  of  these  paths  leads  to  the 
initial  value  of  the  function  after  each  circuit. 

3.  Find  the  connection  of  the  branches  of  the  function  at  the  various 
branch-points.  Having  determined  the  number  of  branches  and 
located  the  branch-points,  the  branches  themselves  are  not  as  yet 
uniquely   determined.     This,   however,   is  not  necessary  in  order 


356  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

that  we  may  determine  the  connection  of  the  branches.  To  show 
this  connection  find  the  cyclic  permutation  of  the  branches  at  each 
branch-point  as  the  independent  variable  describes  a  closed  path  in 
the  ordinary  complex  plane  about  that  point.  The  number  of 
branches  affected  at  any  branch-point  is  one  greater  than  the  order  of 
the  branch-point. 

4.  Draw  the  branch-cuts.  The  branch-cuts  may  be  inserted  in  a 
variety  of  ways.  They  should  be  so  chosen  as  suits  best  the  pur- 
poses of  the  discussion  in  hand.  When  the  cyclic  permutation  of 
the  branches  permits,  the  branch-cuts  may  connect  the  various 
branch-points,  or  when  more  convenient  they  may  be  drawn  from 
the  various  branch-points  to  the  point  at  infinity.  As  we  have  seen, 
a  branch-cut  need  not  be  a  straight  line  and  may  in  fact  be  any 
ordinary  curve  that  does  not  intersect  itself.  When  once  the  branch- 
cuts  are  drawn,  the  various  branches  of  the  function  are  definitely 
determined  aggregates  of  value-pairs  {w,  z),  and  with  each  sheet  of 
the  Riemann  surface  there  is  associated  a  definite  branch  of  the 
function.  The  various  sheets  of  the  surface  should  be  so  connected 
along  the  branch-cuts  that  as  the  variable  passes  over  one  of  these 
cuts  the  proper  branches  of  the  function  interchange. 

While  the  branches  of  a  function  are  single-valued,  it  may  happen 
that  both  w  and  z  are  multiple-valued  functions  of  the  other.  In 
that  case  it  is  often  convenient  to  introduce  a  third  variable  r  so 
related  to  w  and  z  that  the  Riemann  surfaces  constituting  the  W-plane 
and  the  Z-plane,  respectively,  map  continuously  upon  the  single- 
sheeted  T-plane.  Each  branch  of  the  given  function  w  =  f(z)  asso- 
ciated with  a  sheet  of  the  Z-plane  maps  into  a  fundamental  region 
of  the  T-plane,  which  we  may  designate  as  a  (z,  t)  fundamental 
region.  Likewise  each  sheet  of  the  Riemann  surface  constituting 
the  TT-plane  is  associated  with  a  branch  of  the  inverse  function 
2  =  <l>(w)  and  maps  upon  a  fundamental  region  of  the  r-plane,  which 
we  may  designate  as  a  («?,  r)  fundamental  region.  The  (z,  t)  regions 
do  not  coincide  with  the  {w,  t)  regions.  As  a  result,  that  portion  of 
the  Riemann  surface  constituting  the  TT-plane  which  corresponds 
to  a  sheet  of  the  Z-plane,  and  hence  to  a  branch  of  the  function 
"^  =  f{^),  does  not  coincide  exactly  with  the  whole  of  one  or  more 
sheets  of  the  TF-Riemann  surface.  It  may  be  less  or  it  may  be  more 
than  one  sheet,  depending  upon  the  nature  of  the  functional  relation 
between  w  and  z.  We  shall  refer  to  that  portion  of  the  TF-Riemann 
surface  which  corresponds  to  the  whole  of  a  sheet  of  the  Z-plane  as  a 


Art.  65.] 


PROPERTIES  OF  RIEMANN  SURFACES 


357 


fundamental  region  on  the  Riemann  surface.  When  the  branch- 
cuts  are  inserted,  the  correspondence  between  the  points  of  this 
region  and  the  particular  sheet  of  the  Z-pIane  is  definitely  determined; 
that  is,  this  branch  of  the  function  w  =  f{z)  is  fully  determiaed. 
Since  to  a  single  sheet  of  the  Z-plane  there  may  correspond  more 
than  one  sheet  of  the  TF-plane,  it  follows  that  some  of  the  iw-values 
may  be  repeated,  although  the  branches  of  the  given  function  re- 
main single-valued.  An  illustrative  example  will  aid  to  make  clear 
the  foregoing  discussion. 


Ex.  1.   Consider  the  function 


u^  =  ^. 


We  introduce  the  auxiliary  variable  t  by  putting 

The  two-sheeted  Z-Riemann  surface  required  for  this  function  maps  upon  the 
whole  of  the  r-plane,  Fig.  120.  If  we  take  the  negative  half  of  the  axis  of  reals  as 
the  branch-cut,  then  the  upper  sheet  maps  into  the  half  of  the  r-plane  to  the  right 
of  the  axis  of  imaginaries,  while  the  second  sheet  maps  into  the  half  of  the  plane 
to  the  left  of  the  same  axis.    On  the  other  hand,  the  three-sheeted  TT-plane  maps 


T-pl 

ane 

Px 

ZJ' 

i 

/ 

0 

w 

^-            > 

QI 

15     _ 

^, 

\ 

\ 

N 

i" 

:i: 

\P^ 

Fig.  120. 


Fig.  121. 


into  the  whole  of  the  r-plane  as  follows,  where  again  we  take  the  negative  half  of 
the  axis  of  reals  as  the  branch-cut.     The  first  sheet  maps  into  the  region  /  bounded 

by  the  hues  OPx  and  OPi  making  angles  of  ^  and  —  --,  respectively,  with  the 

o  o 

positive  axis  of  reals.  The  other  two  sheets  map  Ukewise  into  regions  //  and 
III,  respectively.  By  direct  comparison  of  the  Z-surface  and  the  TT-surface,  it 
will  be  seen  that  the  region  S  of  the  TT-surface  corresponding  to  the  first  sheet 
of  the  Z-plane  consists  of  the  first  sheet  of  the  W-surface  together  with  the  second 
quadrant  of  the  third  sheet  and  the  third  quadrant  of  the  second  sheet,  Fig.  121. 


358  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

AU  of  the  values  of  S  lying  to  the  left  of  the  axis  of  imaginaries  are  repeated,  as 
will  be  seen  from  the  figure;  for,  that  portion  of  the  region  S  lies  in  two  sheets  of 
the  TF-surface  and  the  one  directly  over  the  other.  However,  no  two  of  these 
values  of  w  correspond  to  the  same  value  of  z;  that  is,  while  some  of  the  values  of  ^ 

the  particular  branch  of  the  given  function  are  repeated  in  S,  nevertheless  the     A  ^ 
branch  is  a  single- valued  function  of  z.  y^*  ^  t/ 

66.  Singular  points  of  multiple-valued  functions.  Since  each  ^ 
branch  of  a  multiple-valued  analytic  function  is  single-valued,  if  we 
exclude  the  branch-points  from  consideration  we  may,  as  we  have 
already  seen,  regard  such  a  function  as  an  aggregate  of  single-valued 
functions,  each  of  which  may  be  holomorphic  for  those  values  of  the 
independent  variable  which  belong  to  the  particular  sheet  of  the 
Riemann  surface  with  which  the  corresponding  branch  is  associated. 
Aside  from  the  branch-points,  each  branch  of  the  given  function 
may  have  such  other  singular  points  as  any  single-valued  function. 
The  singular  points  may  affect  one  sheet  or  more  than  one  sheet  and 
consequently  may  be  singular  points  of  one  or  of  more  than  one 
branch  of  the  given  function. 

For  example,  a  branch  of  a  multiple-valued  function  maj'  have  a  singular 
point  which  does  not  affect  any  other  branch.  As  an  illustration,  consider  the 
function 

w  =  log  log  z.  (1) 

Let  2  =  1.     For  this  value  oiz,w  =  log  z  takes  any  one  of  the  values 
2  kiir,     A;  =  0,  1,  2,  ...  . 

These  values  of  w  correspond  to  the  value  z  =  1  in  the  various  sheets  of  the 
Riemann  surface  constituting  the  Z-plane.  For  the  sheet  corresponding  to 
k  =  0,  w  becomes  infinite  for  z  =  1  and  the  function  (1)  ceases  to  be  regular. 
For  k  r^  Q,  however,  the  function  can  be  expanded  in  powers  of  (z  —  1)  and 
therefore  z  =  1  is  a  regular  point  for  these  sheets.  Consequently,  the  given 
function  has  a  singularity  at  z  =  1,  which,  however,  affects  only  one  branch. 

In  the  neighborhood  of  a  point  Zq  which  is  not  a  branch-point,  a 
multiple-valued  analytic  function  can  be  expanded  in  a  series  in- 
volving only  integral  powers  of  (2  —  Zq).  Such  a  point  is  a  pole  or  an 
essential  singular  point  according  as  the  expansion  contains  a  finite 
or  an  infinite  number  of  terms  having  negative  exponents.  If  there 
are  no  negative  exponents,  then  0o  is  a  regular  point.  Let  us  now 
examine  the  situation  when  20  is  a  branch-point.  Suppose  that  at 
2o  a  finite  number  of  branches,  say  k,  of  the  given  analytic  function 
"^  =  f{z)  are  cyclically  connected.  Take  a  small  region  about  Zq 
bounded  by  a  curve  C  closed  upon  the  Riemann  surface,  such  that 
it  incloses  no  singular  point  nor  branch-point  other  than  Za.    Such 


Art.  66.]  SINGULAR  POINTS  359 

a  curve  must  make  k  circuits  about  Zo  before  it  can  be  said  to  be 
closed.  Let  any  convenient  line  extending  out  indefinitely  from  Zq 
be  taken  as  a  branch-cut.  We  shall  speak  of  the  /c-sheeted  open 
region  thus  obtained  on  the  Riemann  surface,  bounded  by  Zo  and  C, 
as  the  region  R.  Denote  the  function  defined  in  72  by  the  k  branches 
of /(z)  connected  at  zq  by  F(z).    By  means  of  the  substitution 

z-  Zo  =  T^, 

the  region  R  is  mapped  in  a  single-valued  and  continuous  manner 
upon  a  region  S  of  the  one-sheeted  r-plane.  Corresponding  to  the 
point  Zo,  we  have  the  point  r  =  0,  and  except  at  the  point  Zo  itself, 
the  mapping  is  conformal.  The  transformed  function  0(t)  corre- 
sponding to  F{z)  is  single-valued,  and  with  at  most  the  exception  of 
the  point  r  =  0  it  is  holomorphic  in  >S.  At  t  =  0,  the  function 
^(t)  may  have  a  pole  or  an  essential  singularity.  In  either  case,  the 
limit  of  0(t)  as  t  approaches  zero  does  not  exist.  On  the  other  hand 
the  function  <^(t)  may  approach  a  definite  limiting  value  as  r  ap- 
proaches zero.  If  in  the  latter  case  we  assign  this  limiting  value  as 
the  value  of  0(t)  at  t  =  0,  then,  by  virtue  of  Theorem  I,  Art.  51,  the 
origin  is  a  regular  point  of  0(t).  In  any  case  4>(t)  can  be  expanded 
in  the  neighborhood  of  t  =  0  by  means  of  Laurent's  expansion,  thus 
obtaining 

n=m 

which  holds  for  values  of  r  exterior  to  an  arbitrarily  small  circle 
about  the  origin  and  interior  to  any  concentric  circle  within  S.  If 
0(t)  becomes  infinite  as  t  approaches  zero,  then  the  origin  is  a  pole 
and  there  are  a  finite  number  of  negative  terms  in  (2)  equal  in  number 
to  the  order  of  the  pole.  If  the  pole  is  of  order  X,  then  w  =  —  X. 
If  on  the  other  hand  r  =  0  is  an  essential  singular  point,  m  becomes 
—  00 .  If  it  is  a  regular  point,  m  is  equal  to  or  greater  than  zero 
and  the  expansion  reduces  to  a  Taylor  series. 

The  character  of  0(t)  in  the  neighborhood  of  t  =  0  enables  us  to 
determine  the  nature  of  F{z),  and  hence  of  f(z)  for  those  branches 
affected  at  zq.  The  character  of  those  branches  of  f{z)  not  connected 
at  Zo  is  quite  independent  of  the  existence  of  a  branch-point  at  Zo  for 
the  branches  already  considered.  For  those  sheets  not  affected,  this 
point  may  be  a  regular  point  or  a  pole  or  an  essential  singular  point. 
It  may  also  be  a  branch-point  at  which  two  or  more  of  the  remain- 
ing branches  are  associated  with  each  other.     The  expansion  of  F(z) 


360  MULTIPLE-VALUED  FUNCTIONS  (Chap.  VIII. 

in  the  deleted  neighborhood  of  Zo  is  obtained  by  replacing  r  in  (2)  by 
1 

(2  -  2o)*  giving 

00  n 

F(z)  =  X  «"(2  -  ^)^  (3) 

which  holds  for  values  of  z  in  R,  that  is  in  all  of  the  sheets  affected. 
As  we  know  from  Art.  8,  there  are  k,  /c"*  roots  of  (z  —  Zq),  namely: 
111  1 

(Z  -  Zo)*,  C0(Z  -  Zo)*,  W^(2  -  2o)*,    .    •    .    ,  W*-^^  -  2o)*, 

1 
where  (z  —  zo)*  is  now  restricted  to  the  principal  value  of  the  root 
and  CO  is  one  of  the  imaginary  A;"*  roots  of  unity.     To  get  the  expansion 
of  the  individual  branches  of /(z)  composing  F{z),  all  we  need  to  do 
is  to  replace  a„  in  (3)  by 

a„,  a„(o",   .   .   .  ,  a„(co''-^)", 
respectively. 

In  case  t  =  0  is  a  regular  point  of  <j>(t),  then  from  (2)  we  have 

L  4>{t)  =  A, 

r=0 

where  A  is  different  from  zero  or  equal  to  zero  according  as  we  have 
m  =  0  or  m  >  0;  hence,  since  z  approaches  Zo  as  r  approaches  zero, 
we  have 

L  F{z)  =  A. 

Assigning  this  value  as  the  value  of  F{z)  at  Zo,  then  the  branch-point 
Zq  is  called  a  point  of  continuity  of  F{z).  The  expansion  (3)  is  in 
this  case  a  series  of  increasing  positive  fractional  powers  of  (z  —  Zo), 
and  we  have  m  =  0.  If  m  is  greater  than  zero,  say  equal  to  r,  we 
say  that  F{z)  and  hence  /(z)  has  a  zero  point  of  order  r  at  Zo.  The 
function  w  defined  hy  w^  =  z  has  a  point  of  continuity  at  the  branch- 
point z  =  0.  The  expansion  consists  of  one  term,  namely  z^,  and 
the  given  function  has  a  zero  point  of  order  one  at  this  point. 

If  T  =  0  is  a  pole  of  <^(t),  m  is  negative  and  finite.  Let  the  order 
of  the  pole  at  t  =  0  be  r.  Then  the  expansion  (2)  and  therefore 
(3)  has  r  terms  with  negative  exponents.     We  say  that  F{z)  has  a 

pole  of  order  r  at  Zo.     The  function  w  defined  by  the  relation  w^  =  - 

z 

furnishes  an  illustration.  The  point  z  =  0  is  a  branch-point,  and  at 
the  same  time  it  is  a  pole  of  order  one,  the  expansion  consisting  of  a 
single  term  having  a  negative  exponent,  namely  z~^. 


Art.  66.]  SINGULAR  POINTS  361 

If  T  =  0  is  an  essential  singular  point,  then  the  expansion  (2) 
contains  an  infinite  number  of  terms  with  negative  exponents;  that 
is,  we  have  w  =  —  qo  .  In  this  case,  the  transformed  series  (3)  has 
also  an  infinite  number  of  terms  having  negative  fractional  expo- 
nents.   The  point  Zq  is  called  an  essential  singular  point  of  F{z). 

j_ 
The  function  ty  =  e^^  has  such  a  point  at  z  =  0.     This  point  is  a 
simple  branch-point.     The  form  of   the  expansion  in  the  deleted 
neighborhood  of  the  origin  is 

_i       z~        z  * 
w  =  \->rz  ^  +  2T"'"'3T"'"  ■  ■  '  • 

The  origin  is  therefore  an  essential  singular  point  as  well  as  a  branch- 
point. 

Since  the  single-valued  function  0(t)  can  not  have  other  singular 
points  than  poles  and  essential  singular  points,  it  follows  that  the 
foregoing  discussion  exhausts  the  possibiUties  as  regards  the  singu- 
larities of  a  function  at  a  branch-point  when  a  finite  number  of 
sheets  are  connected.  That  is,  a  branch-point  may  also  be  at  the 
same  time  a  point  of  continuity,  a  pole,  or  an  essential  singularity. 
In  any  case  we  shall  class  a  branch-point  among  the  singular  points 
of  a  function,  since  the  expansion  of  the  function  in  the  deleted  neigh- 
borhood of  the  point  involves  fractional  powers  of  the  variable, 
that  is,  the  function  does  not  permit  a  proper  power  series  develop- 
ment. Consequently,  a  branch-point  is  not  to  be  included  in  the 
region  of  existence  of  the  given  function,  and,  moreover,  we  can  not 
reach  such  a  point  by  the  process  of  analytic  continuation  from  any 
regular  point  in  the  Riemann  surface. 

If  an  infinite  number  of  sheets  are  connected  at  a  branch-point, 
then  the  singularity  at  that  point  may  be  of  a  transcendental  char- 
acter.* For  example,  in  the  case  of  the  logarithmic  function,  the 
origin  is  a  singular  point  where  log  z  becomes  infinitely  great  by 
every  possible  approach  of  z  to  the  origin.  It  is  not,  however,  a 
pole  because  the  order  of  the  singularity  is  not  finite.  It  is  in  this 
case  sometimes  spoken  of  as  a  logarithmic  singularity.  It  is  not 
within  the  scope  of  this  volume  to  discuss  the  character  of  the  various 
transcendental  singularities  that  may  occur  at  branch-points,  further 
than  to  point  out  the  illustration  already  cited. 

We  have  excluded  from  the  region  of  existence  of  an  analytic 

*  Cf.  Zoretti,  Lemons  sur  le  prolongement  analytique,  p.  61. 


362  MULTIPLE-VALUED   FUNCTIONS  [Chap.  VIII. 

function  the  poles  and  essential  singular  points  in  the  case  of  single- 
valued  functions,  and  in  the  case  of  multiple-valued  functions  we 
have  excluded  also  branch-points.  It  is  often  convenient  to  con- 
sider the  region  of  existence  as  thus  used,  together  with  those  singu- 
lar points  where  w  and  z  have  a  definite  one-to-one  correspondence. 
Thus  in  single-valued  functions  we  may  include  in  our  consideration 
the  poles  of  a  function,  if  we  associate  with  the  pole  z  =  Zo  the 
functional  value  w  =  oo.  Likewise  in  multiple-valued  functions, 
we  may  include  those  branch-points  at  which  the  function  has  a 
point  of  continuity  or  a  pole,  by  associating  with  the  branch-point 
0  =  00  as  the  corresponding  functional   values,  Wq  =  L  f{z)   and 

10  =  00,  respectively.  When  these  points  of  the  Riemann  surface 
and  their  corresponding  functional  values  are  added  to  the  region 
of  existence  of  an  analytic  function,  we  shall  speak  of  the  result- 
ing aggregate  of  pair-values  (w,  z)  of  the  given  function  as  an 
analjrtic  configuration.  The  essential  singular  points  are  not  in- 
cluded in  the  notion  of  an  analytic  configuration;  for,  corresponding 
to  such  a  singular  point  no  particular  w-point  can  be  associated. 

Just  as  in  the  consideration  of  single-valued  analytic  functions, 
the  singular  points  are  boundary  points  of  the  region  of  existence. 
In  case  of  multiple-valued  analytic  functions,  these  singular  points 
include  the  branch-points  as  well  as  the  poles  and  essential  singular 
points.  When  these  boundary  points  constitute  a  closed  curve 
upon  the  Riemann  surface,  then  the  region  of  existence  of  the  given 
function  has  a  nattiral  boundary;  that  is,  a  boundary  beyond  which 
it  is  impossible  to  proceed  by  analytic  continuations.  The  region 
of  existence  may  be  different  in  different  sheets  of  the  Riemann  sur- 
face for  the  function.  '^ 

67.  Functions  defined  on  a  Riemann  surface.  Physical  appli- 
cations. The  chief  advantage  of  a  Riemann  surface  is  that  it  en- 
ables us  not  only  to  establish  a  one-to-one  correspondence  between 
the  points  of  the  Z-plane  and  those  of  the  TT-plane,  but  also  to  state 
that  any  continuous  path  in  the  one  plane  corresponds  to  a  continu- 
ous path  in  the  other.  Regarding  the  branch-points  as  boundary 
points  of  the  region  of  existence  on  the  Riemann  surface  and  using 
the  term  region  as  in  the  case  of  single-valued  functions  to  mean  an 
aggregate  of  inner  points,  unless  otherwise  stated,  we  can  by  means 
of  Riemann  surfaces  extend  to  multiple-valued  functions  the  general 
properties  already  discussed  for  single-valued  functions.  When  use 
is  made  of  Riemann  surfaces,  the  mapping  by  means  of  multiple- 


Art.  67.]  FUNCTIONS  ON  RIEMANN  SURFACES  363 

valued  analytic  functions  is  conformal  in  any  region  which  contains 
no  branch-points  or  other  singular  points  of  the  function,  provided 
that  the  derivative  of  the  function  is  different  from  zero;  that  is,  in 
any  region  in  which  the  branches  of  the  function  are  holomorphic  and 
their  derivatives  do  not  vanish. 

In  the  neighborhood  of  a  branch-point  Zo  of  finite  order,  the  ex- 
pansion of  a  multiple-valued  function  requires  an  infinite  series 
whose  terms  involve  fractional  powers  of  (z  —  Zo).  In  the  neighbor- 
hood of  any  isolated  singular  point  other  than  a  branch-point,  the 
expansion  of  any  branch  takes  the  form  of  a  Laurent  series.  In  the 
neighborhood  of  a  regular  point,  the  branches  of  the  function  can 
each  be  expanded  in  a  Taylor  series.  The  circle  of  convergence 
may,  however,  lie  partly  in  one  sheet  and  partly  in  another  depend- 
ing upon  the  position  of  the  point  in  whose  neighborhood  the  func- 
tion is  expanded.  In  no  case  can  the  branch-point  he  within  the 
region  of  convergence;  for,  otherwise  there  would  exist  a  regular 
power  series  development  in  the  neighborhood  of  a  branch-point, 
which  we  have  seen  is  impossible.  As  a  consequence  it  will  be 
seen  that  by  the  process  of  analytic  continuation  a  branch-point 
can  not  be  included  in  the  region  of  existence  of  an  anal5rtic  function. 
It  is  for  this  reason,  as  already  stated,  that  we  have  regarded  branch- 
points as  belonging  to  the  boundary  of  the  region  of  existence. 
Having  thus  excluded  the  branch-points,  analytic  continuation  upon 
the  Riemann  surface  can  take  place  along  any  ordinary  curve  just 
as  in  the  case  of  single-valued  functions.  The  path  of  analytic  con- 
tinuation may  he  wholly  in  one  sheet  or  may  pass  from  one  sheet  to 
another  as  the  conditions  require.  Along  any  closed  path  of  ana- 
lytic continuation  the  terminal  value  of  the  function  is  identical 
with  the  initial  value.  If  the  path  incloses  a  branch-point,  then 
it  must  make  as  many  circuits  around  that  point  as  there  are 
sheets  connected  at  the  point  before  the  path  can  be  said  to  be 
closed. 

Similarly  the  process  of  integration  can  be  extended  to  multiple- 
valued  functions,  the  path  of  integration  being  any  continuous 
ordinary  curve  upon  the  Riemann  surface.  In  case  the  path  does 
not  inclose  a  branch-point,  there  is  nothing  new  in  the  process. 
If,  however,  the  path  incloses  a  branch-point,  say  of  order  k,  then 
it  must  pass  k  times  around  that  point  before  the  path  is  closed. 
For  example,  suppose  it  to  be  required  to  integrate  the  function 
It;  =  Vz  along  a  closed  path  C  about  the  origin.     The  closed  path  C 


364  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

can  be  deformed  into  a  double  circle  about  the  origin  without  affect- 
ing the  value  of  the  integral.    We  have  then 

C  t/O  t/O 

• »  r*'     3  0 J,      3  r^'  .  30J,    ^ 

=  ips   I      COS  -^dd  —  p"^  I      sm-^dd  =  0. 

The  residue  of  a  multiple-valued  function  at  an  isolated  singular 
point  is  defined,  as  in  the  case  of  single-valued  functions,  to  be 


Lf/^'^''' 


2 

where  C  is  a  closed  curve  about  the  given  singular  point  and  inclosing 
no  other  singular  point.  In  case  the  point  Zo  is  a  branch-point  at 
which  k  sheets  are  connected,  the  form  of  the  expansion  of  the 
function  is 

*  5 

f(z)  =   2j  ««(2  -  2o)*, 

where  m  =  — Xorw=  — oo  according  as  36  is  a  pole  or  an  essential 
singular  point.     Since  this  series  converges  uniformly,  it  can  be 
integrated  term  by  term,  and  since  C  can  be  taken  as  a  circle  about  Zo,      , 
traversed  k  times,  we  have  as  the  residue  at  zo  fi 

2^J/^)^^  =  2^2;«nj„      iz-Zo)''dz  ,.  I 

=  ka-k,  (  ^       ^' 

where  m  =  —  k.     When  we  have  —  k  <  m  <  0,  the  residue  is  zero. 
To  find  the  residue  when  there  is  an  isolated  singular  point  at 
z  =  oo,  we  have  as  the  expansion  of  the  function 


f(^)=X-n'  ^<0' 


n=m    2* 


and  hence  obtain 


7—.  I  f{z)dz  =  -kak 


It  is  often  necessary  to  employ  multiple-valued  functions  in  map- 
ping from  one  complex  plane  to  another.  In  case  the  inverse 
function  is  single-valued,  we  can  do  as  was  done  in  Chapter  IV, 
namely:  we  can  map  the  whole  of  one  plane  upon  a  portion  of  the 
other  plane,  or  we  can  now  introduce  a  Riemann  surface  in  place  of 


\S^  A 


Art.  67.] 


FUNCTIONS  ON  RIEMANN  SURFACES 


365 


the  one  plane,  thus  making  it  possible  to  map  in  a  continuous  and 
single-valued  manner  the  whole  of  either  plane  upon  the  whole  of 
the  other. 

As  an  illustration,  consider  the  function  w  =  z^.  The  TF-plane 
is  a  double-sheeted  Riemann  surface  having  branch-points  at  w  =  0 
and  w  =  CO.  Let  us  choose  the  negative  half  of  the  axis  of  reals  as 
the  branch-cut.     By  comparing  Figs.  122  and  123,  it  will  be  seen  that 


Fig.  122. 


Fig.  123. 


the  half  of  the  Z-plane  lying  to  the  right  of  the  axis  of  imaginaries 
maps  into  the  first  sheet  of  the  TF-plane  while  the  half  of  that  plane 
to  the  left  of  this  axis  maps  into  the  lower  sheet  of  the  Riemann  sur- 
face constituting  the  PF-plane. 

Any  line  (a)  lying  in  the  right-hand  half  of  the  Z-plane  and  parallel 
to  the  F-axis  maps  into  the  parabola  (A)  lying  wholly  in  the  first 
sheet  of  the  Riemann  surface.  Likewise  any  line  parallel  to  the 
y-axis  and  lying  in  the  left-hand  half  plane  maps  into  a  parabola  lying 
wholly  in  the  second  sheet.  The  line  (6)  parallel  to  the  X-axis  maps 
into  a  parabola  (B)  situated  symmetrically  with  respect  to  the  t/-axis 
and  lying  partly  in  the  first  sheet  and  partly  in  the  second  sheet  as 
indicated,  the  dotted  portion  of  the  curve  indicating  that  part  which 
Hes  in  the  second  sheet.  A  circle  about  the  origin  in  the  Z-plane 
maps  into  a  double  circle  in  the  TT-plane,  one  in  each  sheet.     The 


366  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

two  circles  thus  obtained  constitute  a  closed  path  as  shown  in  Fig.  123. 
The  line  (C)  situated  in  the  first  sheet  and  parallel  to  the  F-axis  maps 
into  the  branch  (c)  of  an  equilateral  hyperbola  lying  in  the  first 
quadrant  of  the  Z-plane.  Similarly,  the  straight  line  (C)  lying  in  the 
second  sheet  directly  underneath  (C)  maps  into  the  branch  (c')  of 
the  same  hyperbola  lying  in  the  third  quadrant.  Lines  parallel  to 
the  U-axis  map  into  hyperbolas  cutting  the  first  hyperbola  at  right 
angles. 

It  is  of  interest  in  this  connection  to  point  out  some  of  the  uses  of 
Riemann  surfaces  in  the  discussion  of  physical  problems.  A  solu- 
tion of  Laplace's  differential  equation  gives  a  potential  function. 
The  solution  may  give  a  multiple-valued  function  u  and  the  corre- 
sponding analytic  function 

w  =  u  -\-  iv  =  f(z) 

to  which  it  gives  rise  may  have  singular  points  other  than  the 
branch-points.  In  the  corresponding  physical  problem,  however, 
the  potential  must  be  single-valued  when  the  boundary  conditions 
are  given.* 

Similarly,  if  we  arrive  at  the  potential  function  through  a  process 
of  integration,  the  function  may  be  a  cyclic  function;  that  is,  it 
may  be  multiple-valued  because  the  path  of  integration  encircles  a 
singular  point  of  the  region  under  consideration.  If  such  points  be 
excluded  from  the  region  by  arbitrarily  small  circles  being  drawn 
about  them,  the  resulting  region  is  multiply  connected.  It  may 
be  made  simply  connected  by  the  introduction  of  barriers  or  cross- 
cuts, in  which  case  the  potential  function  obtained  in  this  manner  is 
single-valued.  If  the  given  region  lies  upon  a  Riemann  surface,  it 
will  be  seen  that  in  order  that  the  potential  shall  be  single-valued  in 
that  region  it  is  necessary  to  restrict  the  region  to  one  sheet  of  the 
Riemann  surface  by  excluding  from  the  region  all  branch-points  and 
branch-cuts. 

Thus  in  Fig.  123,  the  transformation  w  =  z^  enables  us  to  consider  the  potential 
in  an  electrostatic  field  bounded  by  the  parabola  A  and  lying  exterior  to  this 
curve.  This  region  corresponds  to  the  region  in  the  Z-plane  lying  to  the  right 
of  the  line  a  parallel  to  the  F-axis,  Fig.  122.  This  transformation  does  not, 
however,  enable  us  to  consider  a  field  upon  the  Riemann  surface  lying  interior  to 
the  parabola,  since  this  region  contains  the  branch-point  w;  =  0  and  the  negative 
C7-axis,  which  we  have  here  taken  as  a  branch-cut. 

*  See  Jean,  Electricity  and  Magnetism,  Art.  330,  also  Lamb,  Hydrodynamics, 
3d  Ed.,  Art.  62. 


Art.  68.]  FUNCTION  OF  A  FUNCTION  367 

As  another  illustration,  consider  the  function 

z-  1 


w  =  log 


2  +  1 


We  have  already  discussed  the  mapping  of  this  function  where  the  amplitude  6 

z  —  1 
of  T  =  — r-r  is  restricted  to  its  chief  ampUtude.     If  we  mow  allow  0  to  take  all 

2  +  1 

values  we  have  a  multiple-valued  function  and  the  corresponding  Z-plane  must 
consist  of  an  infinite  number  of  sheets.  The  branch-points  of  the  function  are 
located  atz  =  l,z  =  — 1.  All  of  the  sheets  of  the  surface  are  connected  at  these 
two  points,  but  the  cycUc  arrangement  is  in  the  one  case  the  reverse  of  the  other. 
The  two  points  can  then  be  connected  by  a  branch-cut.  Having  excluded  the 
branch-points  by  arbitrarily  small  circles  extending  in  each  sheet  in  succession, 
the  resulting  surface  is  triply  connected.  If  we  now  take  the  branch-cut  as  a 
barrier  or  cross-cut  and  draw  an  additional  cross-cut  from  some  point  on  this 
inner  boundary  of  the  surface  to  the  point  at  infinity,  we  shall  make  the  region 
simply-connected.  Any  region  in  any  sheet  exterior  to  a  curve  inclosing  both  of 
the  two  branch-jwints  may  therefore  be  taken  as  a  suitable  region  for  the  con- 
sideration of  a  potential. 

68.   Function  of  a  function.     We  have  the  following  theorem. 

Theorem.  An  analytic  function  of  an  analytic  function  is  an 
analytic  function. 

We  shall  demonstrate  this  theorem  for  the  case  of  single-valued 
functions.  From  the  discussion  in  this  chapter  it  follows  that  the 
theorem  may  be  extended  to  multiple-valued  analytic  functions. 

Suppose  F(w)  to  be  a  single-valued  analytic  function  of  w,  and 
let  w  =  fiz)  likewise  be  a  single-valued  analytic  function  of  z.  We 
are  to  show  that  F\f{z)\  =  4)(z)  is  also  an  analytic  function  of  z. 
Let  Zo  be  any  regular  point  of  f(z)  such  that  Wo  =  /(^o)  is  a  regular 
point  of  the  function  F{w).  Then  F(w)  can  be  expanded  as  a  power 
series  in  (w  —  Wo) ;  that  is,  we  may  write 

F{w)  =  ao-\-ai(w—Wo)+a2(w—Woy-\-  •  •  •  -\-ak{w-Wo)''+  •  •  •  .  (1) 
Let  the  circle  of  convergence  of  (1)  have  a  radius  equal  to  p.  Since 
w  =  /(z)  is  analytic,  it  follows  that  ak(w  —  Wo)''  is  also  analytic  for 
all  finite  values  of  k  and  hence  may  be  expanded  as  a  power  series  in 
(z  —  Zq).  The  circle  of  convergence  of  these  power  series  may  be 
denoted  by  r,  since  they  have  the  same  radius  of  convergence  for  all 
values  of  k.    We  may  write 

oo  =  /3o.o 

ai(w  -  Wo)   =  /3i.o  +  /3i.i(z  -  Zo)  +  i8i.2(z  -  Zo)^  +  •  •  •  , 

a2(w  -  WoY  =  /32,o  +  /S2.i(z  -  Zo)  +  /32,2(z  -  Zo)2  -I-  •  •  •  , 

az{w  —  WqY  =  /Ss.o  +  /S3,i(z  —  Zo)  +  i83.2(z  -  Zo)^  +  •  •  •  , 


cL"^ 


368  MULTIPLE-VALUED  FUNCTIONS  (Chap.  VIII. 

Substituting  these  values  in  (1),  we  have  the  double  series 
<^(z)  =  |8o,o 

+  ^1.0  +  ^1.1(2  -  2o)  +  /3i.2(z  -  2o)2  +    •    •    • 
+  /32.0  +  /32.i(3  -  2o)  +  182.2(3  -Zoy+    '    '    • 

+  ^3,0  +  Mz  -  2o)  +  M^  -Zoy+  -  ■  ■  .  (2) 


The  rows  of  this  series  converge  absolutely  for  all  of  those  values  of 
2  for  which  \z  —  Zo\=r'<r,  since  each  row  is  a  power  series  with 
the  radius  of  convergence  r.  Summing  by  rows,  the  resulting  series 
each  term  of  which  is  the  sum  of  a  row  in  (2)  is  none  other  than  (1), 
which,  as  we  know,  also  converges  absolutely  for  aU  values  of  w  for 
which  \w  —  Wo\=R<p.  Since  the  series  formed  by  taking  the 
absolute  values  of  the  terms  of  (2)  converges  by  rows,  it  follows  that 
the  double  series  of  these  absolute  values  also  converges  *;  that  is,  the 
given  series  converges  absolutely.  Consequently,  by  the  theorem  of 
Art.  44,  we  may  sum  it  by  colunms  as  well  as  by  rows.     We  have  then 

<t>(z)  =  X  '^*-°  +  X  ^"'^  (2  -  20)  +  X  ^k,2  (2  -  ZoY 
0  1  1 

+  •  •  •  +  2^*-"('^~^)"+  •  •  • ' 
1 

where  we  may  put 

2)/3*.o  ~  ^0,         ^^z^*-"  "^  ^^f        n=  1,  2,  3,  .  .  .  . 
0  1 

The  function  <t>{z)  is  therefore  represented  by  a  power  series  in  the 
neighborhood  of  Zq]  hence,  Zq  is  a  regular  point  of  <t>{z).  But  Zo  is 
any  regular  point  of  w  =  f{z)  for  which  the  corresponding  point  wo 
is  a  regular  point  of  F(w).  Hence,  within  the  region  for  which  z  lies  in 
the  region  of  existence  of  F(w),  we  may  regard  0(2)  as  identical  with 
F\f{z)l.  It  is  possible,  however,  that  the  region  of  existence  of  the 
analytic  function  <f>{z)  thus  defined  may  extend  beyond  the  region  of 
existence  ofw=  f{z) .  On  the  other  hand,  it  is  possible  that  the  values 
of  w  given  by  the  relation  w  =  f(z)  may  not  lie  within  the  region  of  ^ 
existence  of  F{w),  in  which  case  F\f{z)\  has  no  meaning. 

69.  Algebraic  functions.  In  the  preceding  chapter,  we  dis- 
cussed a  special  kind  of  algebraic  functions,  namely  rational  func- 
tions. In  the  present  chapter  we  have  had  occasion  to  consider 
several  particular  algebraic  functions.     We  shall  now  consider  the 

*  See  Bromwich,  Theory  of  Infinite  Series,  Art.  31. 


Art.  69.]  ALGEBRAIC   FUNCTIONS  369 

general  case  where  w  =  f{z)  is  defined  by  an  irreducible  equation  of 

the  form 

F{w,  z)  ^  w-  +/i(0)  «;»-^+/2(z)  w-^  +  •  •  •  +  /„(2)  =  0,  w  >  0,  (1) 

where  /i(z),  Ji{z),  .  .  .  ,  fn{z)  are  rational  functions.*  In  some  dis- 
cussions it  is  convenient  to  write  the  foregoing  equation  in  the  follow- 
ing form 

Vo{z)  w^  +  7>i(z)  w"-^  +  •  •  •  +  Vk{z)  w"-*  +  •  •  •  +  Vn{z)=  0,  (2) 
where  Pfc(z),  A;=  0,  1,  2,  .  .  .  ,  n,  is  a  rational  integral  function  of  2. 
For  each  value  of  z  these  equations  have  n  roots  and  there  are  then 
in  general  n  distinct  values  of  w.  We  shall  denote  these  values  by 
Wi,  Wi,  .  .  .  ,  Wn.  The  function  w  =  /(z)  thus  defined  is  then  a  multi- 
ple-valued function,  and  Wi,  Wi,  .  .  .  ,  Wn  are  all  functions  of  z. 
In  fact  the  functions  Wi,  Wz,  .  .  .  ,  Wn  become  the  n  branches  of  the 
given  function  when  once  the  branch-cuts  are  properly  chosen. 

Theorem  I.  Every  value  of  Zo  for  which  all  of  the  n  branches  of  an 
algebraic  function  remain  finite  and  distinct  is  a  regular  point  of  each 
branch  of  the  function. 

Corresponding  to  a  circle  C  about  Zo  as  a  center  there  may  be 
drawn  a  circle  Ck  in  the  W-plane  about  each  of  the  distinct  points 
Woyk  (A;  =  1,  2,  .  .  .  ,  w)  as  a  center  such  that  for  all  values  of  z  in  the 
region  S  bounded  by  C  each  of  these  circles  Ck  shall  inclose  values  of  w 
belonging  to  one  and  only  one  branch  of  the  given  function.  Conse- 
quently, each  branch  of  the  function  is  single-valued  for  values  of  z 
inS. 

We  shall  now  show  that  Zo  is  a  regular  point  of  each  branch  of  the 
given  algebraic  function.  To  do  this,  it  is  sufiicient  to  show  that  each 
of  the  functions  Wk  (k  =  1,2,  ...,  n)  admits  of  a  derivative  for 
values  of  z  in  S.  Denote  by  Awk  the  increment  of  Wk  corresponding  to 
the  increment'  Az  of  z.     If  the  given  function  is  defined  by  the  equation 

F(w,  z)  =  0, 
we  have 

AF  =  F{w  +  Aiy,  z  +  Az)  -  F{w,  z) 
_  F{w  +  Aw,  z  +  Az)  —  F{w,  z  +  Az) 
Aw 
F(w,  z  +  Az)  -  F{w,  z)  ^^  _  Q 
Az 

*  For  a  somewhat  different  definition  of  an  algebraic  function,  see  Forsyth, 
Theory  of  Functions,  2d  Ed.,  Art.  95.  Compare  also  Encyklopadie  d.  Math.  Wiss., 
Vol.  II,  Bs,  Art.  1. 


370  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

But  since  the  derivatives  -r- ,  -r-    both  exist,   it  follows  that   for 

dw     dz 

w  =  Wk  the  foregoing  equation  can  be  written  in  the  fonn 

where  ti,  C2  approach  zero  with  Awk,  Az,  respectively.  We  have  from 
the  foregoing  relation 

dF 
Awk^_   dz^^ 
Az  dF  ' 

dWk 

As  Az  approaches  zero,  Awk  also  approaches  zero  and  hence  we  have 
in  the  limit  the  same  law  as  holds  for  the  differentiation  of  implicit 
functions  of  real  variables,  namely 

dF 

dWk    _  dz  ,r.s 

dWk 

dF 
The  value  of  - — is  different  from  zero  for  all  values  of  z  in  »S:   for, 
dWk 

otherwise  F(w,  z)  =  0  would  have  a  multiple  root  *  for  some  value  of  z 

in  S,  which  is  contrary  to  the  hypothesis.     The  value  of  -y^  is  given  by 

dz 

(3)  for  any  Wk,  k  =  1,  2,  .  .  .  ,  n.  Consequently,  the  point  2o  is  a 
regular  point  for  each  of  the  functions  Wk  (k  =  1,  2,  .  .  .  ,  n). 

Theorem  II.  The  number  of  points  at  which  two  or  more  of  the 
branches  of  an  algebraic  function  may  become  equal  or  infinite  is  finite. 

The  finite  values  of  z  for  which  two  or  more  of  the  values  Wi, 
W2,  .  .  .  ,  Wn  become  equal  are  those  values  of  z  that  cause  the  dis- 
criminant of  F(w,  z)  =  0  to  vanish.  Consequently,  forming  the 
resultant  t  -B  of  the  two  polynomials 

dF(w,  z) 


F{w,z), 


dw 


the  desired  values  of  z  are  the  roots  of  the  equation  obtained  by 
equating  R  to  zero.  There  can  be  at  most  a  finite  number  of  roots 
of  this  equation. 

The  finite  values  of  z  for  which  two  or  more  of  the  values  wi,  Wi, 

*  See  Forsyth,  Theory  of  Functions,  2d  Ed.,  Art.  94. 
t  See  B6cher,  Introduction  to  Higher  Algebra,  Art.  86. 


Art.  69.]  ALGEBRAIC   FUNCTIONS  371 

.  .  .  ,  Wn  become  infinite  are  those  for  which  the  coefl&cient  po{z) 
of  (2)  vanishes.  Since  po(z)  is  the  least  common  multiple  of  the 
denominators  oifi{z),  f^iz),  .  .  .  ,  f„(z)  and  therefore  of  finite  degree  in 
2,  there  can  be  only  a  finite  number  of  roots  of  the  equation  po{z)  =  0. 

The  only  remaining  z-point  at  which  two  or  more  values  of  Wi, 
W2,  .  .  .  ,Wn  can  become  equal  or  infinite  is  the  point  z  =  oo .  Con- 
sequently, the  total  number  of  points  at  which  the  branches  of  an 
algebraic  function  can  be  infinite  is  finite  in  number,  and  hence  the 
theorem. 

From  Theorem  I  it  follows  that  any  one  of  the  branches  Wk  can 
be  expanded  in  a  Taylor  series 

Wk  =  oo.*  +  ai.fc(2  -  2o)  +  a2.fc(z  -  2o)^  +    '   '   •  ,  (4) 

which  holds  at  least  for  all  values  of  z  in  the  region  bounded  by  the 
circle  of  convergence  C.  The  expansions  for  the  various  branches 
are  of  course  different  and  in  general  the  radius  of  convergence  is 
not  the  same  for  all  branches.  We  now  see  that  there  are  only  a 
finite  number  of  points  at  which  the  branches  of  an  algebraic  func- 
tion may  become  infinite  or  two  or  more  of  them  be  finite  and  equal. 
Since  all  other  points  must  be  regular  points,  it  follows  that  every 
branch  of  an  algebraic  function  is  holomorphic  except  at  a  finite 
number  of  points,  and  hence  we  have  the  following  theorem. 

Theorem  III.  Every  algebraic  function  is  analytic  and  has  only  a 
finite  number  of  singular  points. 

The  expansion  (4)  of  any  branch  Wk  in  the  neighborhood  of  a  point 
where  that  branch  is  finite  and  distinct  defines  an  element  of  the 
function,  and  from  this  element  the  algebraic  function  is  completely 
and  uniquely  determined.  It  is  also  of  interest  to  note  that  it  fol- 
lows from  Theorem  I  that  the  singularities  of  an  algebraic  function 
can  occur  only  at  points  where  two  or  more  of  the  branches  have  the 
same  finite  value  or  where  one  or  more  of  the  branches  become 
infinite. 

We  shall  use  the  expression  infinity  of  a  function,  or  more  briefly 
an  infinity,  to  mean  a  singular  point  of  a  multiple-valued  function 
at  which  the  function  becomes  infinite  by  at  least  one  approach  of 
the  independent  variable  to  the  critical  point.  Infinities  include 
both  poles  and  essential  singular  points  but  exclude  branch-points, 
unless  those  points  are  at  the  same  time  poles  or  essential  singular 
points.     The  order  of  an  infinity  may  therefore  be  either  finite  or 


372  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

infinite.     By  the  following  theorem  we  shall  show  that  the  infinities 
of  an  algebraic  function  are  necessarily  poles. 

Theorem  IV.  The  infinities  of  an  algebraic  function  are  singular 
points  of  the  coefficients  and  conversely.  Moreover,  an  algebraic  func- 
tion can  have  only  polar  infinities. 

Let  the  given  function  w  =  f{z)  be  defined  by  the  algebraic  equa- 
tion 

F(w,  z)  ^  w-  +fi{z)w--'  +  '  '  '  +fn(z)  =  0.  (5) 

We  shall  first  show  that  if  2o  is  an  infinity  of  any  branch  Wk,  it  is 
at  the  same  time  a  singular  point  of  at  least  one  of  the  coefficients 
/i>  fiy  •  '  •  ,  fn  of  (5)  and  conversely.  We  shall  also  show  that  the 
order  of  the  singularity  of  this  coefficient  is  equal  to  or  greater  than 
the  order  of  the  infinity  of  Wk.  From  (5)  we  have  by  aid  of  the 
relation  between  the  roots  and  coefficients  of  an  algebraic  equation 

fl{z)   =    -  iWi  +  W2-\-    •    •    •    +Wk+    •    •    •    +  Wn), 

fi{z)   =  WiWi  +  W1W3  +    •    •    •    +  WkW2  +  WkWs  +    •    •    •    +  Wn-lWn) 

fn(z)   =   {-l)''WiW2W3    .    .    .    Wk    .    .    .    Wn. 

Let  the  branches  of  the  given  function,  other  than  Wk,  be  determined 
by  an  equation  of  the  form 

W"-'  +  <t>liz)  W^-^  +  02(2:)  «;"-3  +    .    .    .    4-  0„_2(2)  w  +  <l>n-i(z)   =  0.     (6) 

Then  we  may  write 

fi(z)  =  -Wk  +  Mz), 
/2(z)  =  —  Wk<i>i{z)  +  (h{z), 


(7) 


fn-l{z)   =    -  Wk<i>n-i{z)  +  <t>n-l{z), 
fn{z)   =    -  Wk<t>n-l{z). 

From  the  last  of  equations  (7)  it  follows  that  zq  must  be  a  singular 
point  of  fn{z)  unless  (t>n-i{z)  has  a  zero  point  of  as  high  order  as  the 
infinity  of  Wk.  The  singularity  of  fn{z)  may,  however,  be  of  higher 
order  than  the  infinity  of  Wk  in  case  20  is  also  an  infinity  of  <l>n-\{z). 
It  may  be  of  lower  order  provided  0„-](z)  has  a  zero  point  at  Zo  of 
order  less  than  the  order  of  the  infinity  of  Wk.  Consequently,  fn{z) 
has  a  singular  point  at  Zq  of  order  equal  to  or  higher  than  the  infinity 
of  Wk  unless  20  is  at  the  same  time  a  zero  point  of  </>„_!  (2).  In  the 
latter  case,  it  follows  in  a  similar  manner  from  the  next  to  the  last 
of  these  equations  that  fn-\{z)  has  a  singularity  at  Zq  of  order  equal 


Art.  69.]  ALGEBRAIC  FUNCTIONS  373 

to  or  higher  than  the  infinity  of  Wk  unless  <i)n-2{z)  has  a  zero  point 
at  Zo-  Continuing  in  this  manner,  it  follows  that  either  some  one 
of  the  coefficients  fiiz),  .  .  .  ,  Jn{z)  has  an  infinity  at  Zq  of  at  least 
as  high  an  order  as  that  of  Wky  or  ^i{z)  has  a  zero  point  at  Zq.  In  the 
latter  case,  it  follows  from  the  first  equation  that  fi{z)  has  an  infinity 
at  Zq  of  the  same  order  as  Wk.  We  can  conclude  that  at  least  one  of 
the  coefficients  /i(z),  U{z),  •  •  •  ,  fniz)  has  a  singularity  at  Zo  of 
equal  or  higher  order  than  the  infinity  of  Wk.  However,  the  coefii- 
cients  Ji{z),  fi{z),  .  .  .  ,  fn(z)  are  all  rational  functions  of  z  and  there- 
fore the  function  itself  can  have  at  most  polar  infinities. 

Conversely,  if  some  one  of  the  coefficients  fi(z),  fiiz),  .  .  .  ,  /„(z) 
has  a  singular  point  at  Zo,  then  it  must  be  a  pole.  From  (7)  it  fol- 
lows that  if  zo  is  not  a  pole  of  Wk,  it  must  be  a  pole  of  at  least  one  of 
the  coefiicients  0i(z),  <h(z),  •  .  .  ,  0n-i(2)  of  (6).  Then  by  similar 
reasoning  Zq  is  a  pole  of  some  branch,  say  Wk',  determined  by  (6)  or 
of  at  least  one  of  the  coefiicients  of  the  resulting  equation  after  both 
of  the  branches  Wk  and  Wh'  have  been  removed.  Continuing  in  this 
manner,  it  follows  that  either  Zo  is  a  pole  of  some  one  of  the  first 
n  —  1  branches  or  of  the  coefficient  in  the  last  equation  and  hence 
of  the  last  of  the  branches. 

The  poles  which  appear  as  the  singular  points  of  an  algebraic 
function  may  at  the  same  time  be  branch-points,  but  the  function 
can  have  no  other  singularities  than  poles  and  branch-points,  and 
as  we  have  seen,  an  algebraic  function  can  have  in  all  at  most  a 
finite  number  of  singular  points.  If  the  branch-point  is  at  the  same 
time  a  pole,  the  expansion  of  the  function  in  an  infinite  series  for 
values  of  z  in  the  neighborhood  of  that  point  involves  a  finite  number 
of  terms  having  negative  fractional  exponents.  If  the  pole  occurs 
at  a  point  other  than  a  branch-point  the  expansion  of  any  branch  of 
the  function  involves  a  finite  number  of  negative  integral  exponents. 
In  the  neighborhood  of  a  branch-point  which  is  not  a  pole  the  ex- 
pansion involves  only  positive  fractional  exponents.  In  the  neigh- 
borhood of  any  other  finite  point  the  expansion  of  each  branch  is 
accomplished  by  an  ordinary  Taylor  series.     The  expansion  of  the 

function  in  the  neighborhood  of  z  =  oo  is  in  terms  of  - ,  the  character 

of  the  expansion  depending  upon  the  nature  of  the  function  at  that 
point. 

We  shall  now  consider  whether  every  function  /(z)  having  no  sin- 
gular points  other  than  poles  and  branch-points  must  be  an  alge- 


374  MULTIPLE- VALUED  FUNCTIONS  [Chap.  VIIL 

braic  function.  Before  discussing  that  question,  however,  we  must 
demonstrate  the  following  proposition. 

Theorem  V.  //  in  a  given  region  S  of  the  complex  plane,  a  func- 
tion w  =  f{z)  ha^  no  singular  points  other  than  poles  and  branch-points 
and  for  values  of  z  in  S  has  m  branches,  which  are  in  general  distinct, 
then  any  symmetric  polynomial  of  these  branches  is  in  S  a  mero- 
mxyrphic  function  of  z. 

Denote  the  m  branches  of  w  by  Wi,  w^,  .  .  .  ,  Wm.  For  any  value  of 
z  these  branches  of  w  may  remain  distinct,  or  two  or  more  of  them 
may  take  the  same  finite  value,  or  finally  one  or  more  of  them  may 
become  infinite.  These  branches  are  m  single-valued  functions  of 
z  when  we  consider  the  branch-cuts  as  replaced  by  barriers  restrict- 
ing the  variation  of  z.  From  the  given  hypothesis  it  follows  that 
each  Wk  admits  of  a  derivative  except  at  certain  singular  points,  and 
hence  in  that  portion  of  the  region  S  exclusive  of  singular  points  Wk 
is  holomorphic. 

The  expansion  of  Wk  in  the  neighborhood  of  any  point  Zo  which  is 
not  a  pole  or  a  branch-point  is  of  the  form 

Wk  =  ao-h  0:1(2  -  2o)  +  0:2(3  -  Zo)^  +  •  •  •  +  an(2  -  2o)"+  •  •  •  .   (8) 

In  case  Zo  is  a  branch-point  of  the  function  lying  in  S,  where  r 
values  of  w  become  equal  but  remain  finite,  the  expansion  of  the 
given  function  for  those  branches  is  obtained  by  introducing  the 
auxiliary  function 

z—  Zo  =  T^ 

As  we  have  seen,  this  substitution  leads  to  an  expansion  of  the  given 
function  in  the  neighborhood  of  Zo  of  the  form 

12  n 

oo  +  oci{z  -  ZoY  -h  a2{z  -  ZoY  +   •  •  •   +  a„(z  -  ZqY -\-   ■  ■  -  .     (9) 

No  terms  with  negative  exponents  appear  in  the  expansion  since  the 
branch-point  is  not  an  infinity.  To  get  the  expansion  in  the  neigh- 
borhood of  Zo  of  each  of  the  r  branches  associated  at  that  point, 

n 

we  need  to  regard  (z  —  ZoY,  n  =  1,2,3,  .  .  .  ,  as  the  principal  value 
of  the  r""  root  of  (z  —  20)"  and  replace  an  by 

a„,  anO}"",  anio)^y,  .  .  .  ,  a„(co'-0", 

where  «  is  one  of  the  r  imaginary  r'^  roots  of  unity,  as  explained  in 
Art.  66.  The  remaining  m  —  r  branches  may  remain  distinct  or 
form  by  themselves  one  or  more  cycles. 


Art.  69.]  ALGEBRAIC   FUNCTIONS  375 

Any  symmetric  polynomial  of  the  m  branches  wi,  w^,  .  .  .  ,  Wm 
can  be  expressed  in  terms  of  the  sums  of  equal  powers  of  the  Wks; 
that  is,  in  terms  of  functions  of  the  type  * 

In  adding  equal  powers  of  WkS,  however,  the  coefficients  of  terms 
having  fractional  exponents  vanish  by  virtue  of  the  relation 

1  +  w  +  aj2  +    .  .   •   +  oi'-^  =  0. 

Hence,  in  this  case  also,  the  expansion  of  any  symmetric  polynomial 
of  Wi,  W2,  .  .  .  ,  Wm  involves  only  positive  integral  powers  of 
(z  —  Zo),  and  consequently  Zq  is  a  regular  point  of  the  given  function 

Finally,  let  us  suppose  Zo  is  a  pole  of  the  given  function.  In  this 
point  then  one  or  more  values  of  Wk  become  infinite  as  z  approaches 
Zq.  In  the  latter  case  the  point  Zq  may  be  a  branch-point.  The 
expansion  of  Wk  in  the  neighborhood  of  such  a  point  involves  a  finite 
number  of  terms  with  negative  exponents,  which  may  be  either  frac- 
tional or  integral.  In  forming  a  symmetric  function  of  Wi,  w^,  .  .  .  , 
Wm  the  exponents  all  become  integral  even  in  case  20  is  a  branch-point 
and  hence  the  resulting  function  has  a  pole  at  Zq. 

As  we  now  see,  any  symmetric  function  of  W\,  W2,  .  .  .  ,  Wm  can 
have  in  the  region  *S  only  polar  singularities.  There  can  be  but 
a  finite  number  of  poles  in  S,  for  otherwise  there  must  be  at  least 
one  essential  singular  point.  Consequently,  any  symmetric  func- 
tion of  Wi,  Wo,  .  .  ,  ,  Wm  must  be  meromorphic  in  the  given  region 
S  as  the  theorem  requires. 

We  shall  now  consider  the  following  theorem. 

Theorem  VI.  Every  analytic  function  w  =  f(z)  having  n  values  for 
each  value  of  z  and  having  in  the  entire  complex  plane  no  other  singu- 
larities than  poles  and  branch-points  can  be  expressed  as  a  root  of  an 
algebraic  equation  of  degree  n  in  w,  the  coefficients  of  which  are  rational 
functions  of  z,  and  consequently  w  =  /(z)  is  an  algebraic  function. 

Corresponding  to  any  point  zo  of  the  complex  plane,  it  follows  from 
the  hypothesis  set  forth  in  the  theorem  that  w  has  n  values,  which  as 
before  we  denote  by  Wi,  W2,  .  .  .  ,  ty„.  These  values  are  in  general 
distinct. 

*  See  Bocher,  Introduction  to  Higher  Algebra,  p.  241. 


376  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

We  shall  now  consider  the  following  symmetric  polynomials  of 

Wl,  V)2,    .    .    .   ,   Wn. 

/l(z)    =    -  (t«l  +  M?2  +     •    •    •    +  Wn), 
ftiz)   =  WiUh  4-  W1W3  +    •    •    •    +  Wn-lW„, 


Mz)   =   (-l)"W?lW2    .    .    .    Wn. 

From  Theorem  V,  it  follows  that  the  functions  fi{z),  f^iz),  .  .  .  , 
Sn{z)  are  meroraorphic  for  all  values  of  z,  since  the  region  S  may  now 
include  the  whole  complex  plane.  These  functions  therefore  have 
only  polar  singularities  and  must  be  rational  functions. 

To  complete  the  proof,  we  have  only  to  equate  to  zero  the  product 

n 

JJ"  {w  —  Wk),  namely 

{w  —  Wi)  (w  —  W2)   .  .   .   (w  —  Wn)  =  0.  (10) 

By  the  principles  of  elementary  algebra,  this  equation  can  be  written 
in  the  form 

w^+Mz)  w--'  +Mz)  w;"-2  +  •  •  •  +fn{z)  =  0. 

The  function  w  =  f(z)  is  a  root  of  this  equation,  and  as  we  have  seen 
the  coefficients /i(2),/2(3),  .  .  .  , /n(z)  are  rational  functions.  Hence 
the  theorem. 

The  Riemann  surface  for  an  algebraic  function  can  now  be  de- 
termined. If  the  equation  F{w,  z)  =  0  defining  the  function  is  of 
the  degree  n  in  w  then  the  Z-plane  must  be  an  n-sheeted  surface. 
The  branch-points  and  the  connection  of  the  sheets  at  each  can  be 
determined  by  the  methods  given.  The  branch-cuts  can  be  drawn 
between  branch-points  where  the  same  sheets  are  affected  but  their 
cyclic  arrangements  are  of  opposite  order,  or  they  may  be  drawn 
to  the  point  z  =  <xi.  All  of  the  sheets  must  form  a  connected  sur- 
face since  by  definition  F{w,  2)  =  0  is  an  irreducible  equation.  In 
case  this  equation  is  reducible,  that  is,  can  be  separated  into  two  or 
more  factors  involving  w  and  z,  then  it  defines  not  one  algebraic 
function  but  two  or  more  such  functions  and  the  corresponding 
Riemann  surface  separates  into  two  or  more  distinct  surfaces.  In 
fact,  a  given  relation  between  w  and  z  may  be  said  to  represent  one 
multiple-valued  function  or  more  than  one  function  according  as 
the  corresponding  Riemann  surface  consists  of  one  connected  part 
or  of  two  distinct  parts. 


Akt.  69.]  EXERCISES  377 

EXERCISES 

1.   Show  that  z",  where  a  =  a  +ib,issi  multiple- valued  analytic  function  with 
a  branch-point  of  an  infinitely  high  order  at  the  origin. 
hint:  Puts"  =  e"''»8^ 

i>    2.   Given  the  analytic  function  w  =     ,  Locate  the  branch-points 

V2  +z2 

and  determine  whether  they  are  at  the  same  time  poles. 

J'        dz 
,  where  C  is  a  curve  closed  on  the 
cVz^  +  1 

Riemann  surface  about  the  point  z  =  i. 
A    4.   Knowing  that 
U  c     dz 

arc  tan  2  =   I    -r—. — ,, 
^0    1  -f-  2* 

expand  in  an  infinite  series  the  function  }{z)  =  arc  tan  z.  How  large  is  the  circle 
of  convergence  of  this  series?  What  singular  points  restrict  the  size  of  the  circle 
of  convergence?     Are  these  singular  points  also  branch-points? 

6.  Determine  the  branch-points  of  the  fimction  w  =  arc  sin  z. 
^  6.   By  aid  of  the  definitions  of  circular  functions,  show  that 

w  =  arc  tan  z  =  tt--  log  -z :-  •     /f'^  ly^  ^ 

2i     °  1  —  zz      '         ' 

Locate  the  branch-points  of  w  and  determine  a  region  on  the  Riemann  surface 
in  which  each  of  the  infinite  number  of  branches  is  holomorphic.  Show  whether 
this  region  is  simply  or  multiply  connected. 

7.  Show  that  the  function  /(z)  =  log  (sin  z)  is  analytic. 

8.  Discuss  the  Riemann  surface  for  the  function  w  =  y/  {z  —  a.)  {z  —  ^8)*. 
What  physical  phenomena  does  this  functional  relation  represent? 

9;   Construct  a  model  showing  the  connection  of  the  sheets  of  the  Riemann 
surface  required  for  the  function  discussed  in  Art.  61. 

10.  Discuss  the  Riemann  surface  for  the  function  "»?■  —  1  =  z'.  Determine  a 
fundamental  region  on  the  TF-Riemann  surface.  ^~'^t.^h\     ct 

11.  By  mapping  the  Z-plane  upon  the  TF-plane  by  means  of  the  function 
w  =  e",  show  that  w  takes  every  value,  except  zero  and  infinity,  in  the  neighbor- 
hood of  z  =  00  . 

12.  Given  the  algebraic  function,  i^  —  18iy  —  35z  =  0.  Determine  the 
character  of  the  Riemann  surface  suited  to  this  function  by  the  method  of  Art.  60. 

L*   13.   Expand  the  function  w  =  Vz  in  an  infinite  series  for  values  of  z  in  the 

.ir 

neighborhood  of  z  =  a,  where  o  =  2  c  3. 
o^    14.   Given  the  function  w  =  Vl  —  z^.    Examine  the  character  of  this  func- 
tion in  the  neighborhood  of  z  =  oo . 
^    15.   Indicate  the  form  of  the  expansion  in  an  infinite  series  which  the  analytic 
function  w  =  f{z)  takes   for   values  of  z  in  the  neighborhood  of  z  =  i,  (a)  if 
z  =  i  is  a  point  of  continuity  and  a  branch-point  of  order  2;  (6)  if  z  =  i  is  a  pole 
of  order  k  but  not  a  branch-point;     (c)  if  z  =  i  is  a  branch-point  where  3  branches  f 
come  together  and  at  the  same  time  a  pole  of  order  4;   (d)  if  z  =  i  is  a  branch- 
point of  order  6  and  an  essential  singular  point;    (e)  if  z  =  i  is  a  zero-point  of 
order  3  and  a  branch-point  where  2  branches  become  equal. 


/ 


378  MULTIPLE-VALUED  FUNCTIONS  [Chap.  VIII. 

16.  Distinguish  between  an  algebraic  function  and  a  transcendental  function 
(o)  as  to  the  number  of  possible  zero  points,  (b)  as  to  the  number  of  poles  and 
essential  singular  points  that  may  occur,  (c)  as  to  the  number  and  order  of  the 
branch-points  that  the  function  may  have.  Illustrate  in  each  case  by  a  particu- 
lar function. 

17.  Does  the  expression  arc  sin  (cos  z)  represent  a  multiple-valued  analytic 
function  or  a  number  of  single-valued  analytic  functions? 

18.  Discuss  the  Riemann  surfaces  for  the  following  functions: 


(o)  w  =  Vz  —  i  +  Vz  —  2, 
(6)  u,  =  VF^l  +  ^=' 
(c)    10*  —  4w  =  z. 


INDEX 


INDEX 


(Numbers  refer  to  pages.) 


Absolute  convergence  of  infinite  prod- 
ucts, 311. 
Absolute  convergence  of  series,  201. 
theorems  concerning,  201,  203,  204, 
206,  208,  212,  214,  227,  228,  236. 
Addition  of  complex  numbers,  8. 

of  infinite  series,  206. 
Algebraic  functions,  23,  368. 

theorems  concerning,  369,  370,  371, 
372,  375. 
Amplitude,  definition  of,  6. 
Analytic  configuration,  362. 
Analytic  curve,  253. 
Analytic  continuation,  245,  249. 
by  power  series,  251. 
by  Schwarz's  method,  252. 
theorems  concerning,  250,  255. 
Analytic  function,  definition  of,  45,  257. 

of  an  analytic  function,  367. 
Anharmonic  ratio,  178. 
Applications  to  physics,  96,  103,   104, 
107,  112,  113,    117,  118,  119,  120, 
132,  137,  138,  139,  140,  141,    152, 
190,  196,  362,  366. 

Boundary  point,  20,  348. 
Bounded  sequence,  28. 
Branch  of  a  function,  332,  355. 

theorems  concerning,  369,  370,  374. 
Branch-cuts,  338,  347,  356. 

theorems  concerning,  351,  352. 
Branch-point,  333,  347,  355. 

order  of,  334. 

theorems  concerning,  335,  351,  352, 
353,  374,  375. 

Cauchy-Goursat  theorem,  66,  68. 

extension  of,  71. 
Cauchy's  integral  formula,  75. 


Cauchy-Riemann  differential  equations, 

83. 
Change  of  variable,  64,  89. 
Chief  amplitude  of  a  complex  number,  7. 
Circle  of  convergence,  definition  of,  230. 

theorems  concerning,  235,  236,  237, 
274. 
Class  of  a  function,  316. 
Closed  region,  20. 
Complex  number,  modulus  of,  6. 

amplitude  of,  6. 

chief  amplitude  of,  7. 
Complex  numbers,  definition  of,  5. 

addition  and  subtraction  of,  8. 

comparison  of,  8. 

division  of,  16. 

geometric  representation  of,  6. 

multiplication  of,  11. 
Complex  plane,  6. 

Conditional  convergence  of  series,  defi- 
nition of,  204. 

theorems  concerning,  205,  237. 
Conditional    convergence    of    infinite 

products,  311. 
Conformal  mapping,  definitions  of,  107. 

theorems  concerning,  107,  175,  189. 
Conjugate  functions,  definition  of,  101. 
Conjugate  points,  definition  of,  170. 

theorems  concerning,  172,  173,  174. 
Continuity,  definition  of,  33. 

theorems  concerning,  34,  35,  38,  39, 
40. 

point  of,  360. 
Continuity  of  f'{z),  77. 
Convergence  of  infinite  series,  defini- 
tion of,  198,  213. 

theorems  concerning,  199,  200,  201, 
207,  208,  214. 
Cross-cut,  53. 


381 


382 


INDEX 


D^jree  of  a  function,  22. 
Deleted  neighborhood,  20. 
DeMoivre's  theorem,  12. 
Derivative,  definition  of,  43. 

continuity  of,  77. 
Derived  function  /'(z),  prof>erties   of, 

77. 
Differentiation  of  series,  222,  225,  237. 

under  integral  sign,  52. 
Difference  of  two  series,  208. 
Division  of  complex  numbers,  16. 
Double  series,  213. 

Doubly  periodic  functions,   318,   319, 
324,  325,  326. 

Element  of  an  analytic  function,  250. 
Electric  potential,  96. 
Elliptic  motion,  194. 
Equianharmonic  points,  182. 
Equipotential  lines,  104. 
Equipotential  surface,  103. 
Essential  singular  point,  263,  273. 

theorems  concerning,  270,  271,  292. 
Expansion,  160. 

modulus  of,  160. 
Expansion  of  functions  in  series,  238. 

Function,  definition  of,  21. 

analytic  function,  45,  257. 
Functions,  definition  of  elementary, 

exponential,  122. 

hyperbolic,  150. 

logarithmic,  133. 

trigonometric,  144. 
Function  of  a  function,  367. 
Fundamental  region,  116,  357. 
Fundamental  theorem  of  Algebra,  291. 

Geometric  inversion,  166. 

of  straight  line,  168. 

of  circle,  168. 

effect  on  angles,  169. 

theorems  concerning,  171,  172,  173. 
Green's  theorem,  54. 

Harmonic  points,  181. 
Higher  derivatives  of  /  («),  79. 


Holomorphic  in  a  region,  45. 
theorems  concerning,  45,  66,  68,  71, 
74,  75,  77,  80,  82,  83,  85,  92,  93, 
224,  235,  238,  247,  248,  250,  263, 
265,  268,  273,  278,  284,  285,  296, 
304,  323. 

Hyperbolic  functions,  150. 

Hyperbolic  motion,  193. 

Indefinite  integrals,  90. 

Infinite  product,  definition  of,  308. 

absolute  convergence  of,  311. 

theorems  concerning,  309,  311. 
Infinity,  point  at,  157. 
Infinity  of  a  function,  371. 

theorem  concerning,  372. 
Inner  point,  20. 

Integral,  positive  direction  of,  52,  63. 
Integral  of/  (2),  definition  of,  60. 

theorems  concerning,  62,  63,  64,  72, 
73,  74,  80. 
Integral  of  /  (z),  when  independent  of 

path,  73. 
Integral,  function  /  (2)  defined  by,  80. 
Integration  of  series,  222,  224,  237. 
Inverse  functions,  property  of,  85. 
Irrational  function,  23. 
Irrational  numbers,  2. 
Isogonal  mapping,  107. 

theorem  concerning,  107. 
Isolated  singular  point,  263. 
Isolated  line  of  singularities,  265. 

Lacunary  space,  258. 
Laplace's  equation,  92. 

consequence  of,  93. 
Level  lines,  103. 
Laurent's  series,  278. 

function    uniquely    determined    by, 
280. 

principal  part  of,  279. 
Limits,  23. 
Limit  of  a  function,  26. 

theorem  concerning,  27. 
Limit  of  a  sequence,  24. 

theorems  concerning,  24,  29,  30,  31. 
Linear  automorphic  functions,  114. 


INDEX 


383 


Linear  fractional  transformations,  156. 

classifications  of,  190. 

general  properties  of,  173. 
Line-integral,  definition  of,  46. 

theorems  concerning,  51,  52,  54,  56, 
57,  58. 
Lines  of  force,  104. 

of  slope,  103. 

of  level,  103. 
Liouville's  theorem,  274. 
Logarithm,  definition  of,  133. 

principal  value  of,  135. 
Logarithmic  derivative,   definition  of, 
296. 

properties  of,  296,  297. 
Logarithmic  potential,  96. 
Logarithmic  spiral  motion,  161. 
Loxodromic  motion,  194. 
Lower  limit,  28. 

Maclanrin's  series,  239. 
Map  oiw=  2^,  105. 

w  =  z",  114. 

2=u;+e«',  13L 

u,  =  log^    138. 
z+  1 

w  =  log  (z  +  1)  (2  -  1),  139. 

(z  -  ly 

{z+  1) 

w  =  COS  z,  147. 

w  =  coshz,  151. 

the  anharmonic  ratios,  181. 
Mapping  in  neighborhood  of  a  regular 

point,  107. 
Maximum,  28. 
Mercator's  projection,  137. 
Meromorphic  in  a  region,  definition  of, 
293. 

theorem  concerning,  374. 
Minimum,  28. 

Mittag-Leffler's  theorem,  303. 
Modulus,  definition  of,  6. 
Moduli,  relation  between,  9,  10. 
Modulus  of  expansion,  160. 
Monogenic  anal3i,ic  function,  258. 
Morera's  theorem,  80. 
Multiplication  of  complex  numbers,  11. 


Multiply  connected  regions,  53. 

integral  over  boundary  of,  74. 
Multiple-valued  function,  definition  of, 
22. 

applications  of,  366. 

illustrations  of,  329. 

Natural  boundary,  258,  362, 
Neighborhood,  20. 
Net  of  points,  321. 
Non-essential  singular  point,  262. 
Norm  of  a  complex  number,  7. 

Open  region,  20. 
Order  of  a  pole,  262. 
a  zero  point,  266. 
Ordinary  curve,  definition  of,  47. 

Parabolic  motion,  191. 

Painlev6's  theorem,  250. 

Partition,  definition  of,  3. 

Path  of  integration,  definition  of,  47. 

deformation  of,  73. 
Periodic  function,  125,  317. 

theorems  concerning,  319,  323,  324, 
325,  326. 
Period-parallelogram,  321. 
Period  region,  322,  323,  326. 
Plane  doublet,  118. 
Point  of  continuity,  360. 
Point  of  equilibrium,  141,  143. 
Pole,  definition  of,  262,  273,  360. 

order  of,  262,  360. 
Poles,  theorems  concerning,  267,  268, 
270,  284,  285,  286,  290,  292,  294, 
295,  296,  297,  298,  325,  326,  335, 
372,  374,  375. 
Potential,  definition  of,  96,  98. 
Power  series,  226. 

theorems  concerning,  227,  228,  230, 
232,  235,  236,  237,  238,  274. 

with  negative  exponents,  275. 
Primitive  element  of  an  analytic  fimc- 

tion,  258. 
Primitive  period,  126. 
Primitive  period  pair,  318. 
Product  of  series,  208. 
Products,  infinite,  308. 


384 


INDEX 


Quotient  of  two  series,  212. 

Radius  of  convergence,  230,  232. 
Ratio  of  magnification,  109. 
Rational  numbers,  1. 
Rational  functions,  290. 

theorems  concerning,  292,  294,  295, 
298,  299. 
Rational  integral  function,  22. 

theorems  concerning,  290,  291. 
Rational  fractional  function,  22. 
Real  numbers,  system  of,  4. 
Reciprocation,  167. 
Reflection,  167,  254. 
R^on,  definition  of,  20. 

of  convergence  of  a  series,  218. 

of  existence,  258,  362. 

of  f>eriodicity,  125. 
R^ular  F)oint,  definition  of,  45. 

theorems  concerning,  263,  265,  369. 
Residue,  definition  of,  284,  364. 

theorems  concerning,  284,  285,  286, 
295,  296,  297,  325. 
Rieraann's  theorem,  263. 
Riemann  surface  for  w  =  -s/z,  336. 

ioxu^  —  Zw—2z  =0,  338. 

8  I ^ 

iox  w=^  z-z^-\-\l  — — ,  343. 

"     2   —    Zi 

for  tc  =  log  z,  346. 
Riemann  surfaces,  properties  of,  355. 

functions  defined  on,  362. 

mapping  on,  364. 
Roots  of  a  complex  number,  13-16. 
Root,  principal  value  of,  14. 

Sequence,  limit  of,  24. 

theorems  concerning,  24,  31. 
Sequence  of  circles,  limit  of,  29. 
Sequence  of  rectangles,  limit  of,  30. 
Series,  convergence  of,  198,  199,  200, 
201. 

differentiation  of,  222. 

double,  213. 

integration  of,  222. 

operations  with,  206. 

power,  226. 

imiform  convergence  of,  217. 


Simple  pole,  263. 

Simply  connected  region,  definition  of, 

52. 
Simply  periodic  functions,  126. 
Single-valued  function,  22,  245. 
Singular  point,  45,  262,  358. 
Sink,  118. 
Source,  118. 

Stereographic  projection,  184,  354. 
Subtraction  of  complex  numbers,  10, 
Sum  of  two  series,  206. 

Taylor's  series,  239. 

Trigonometric  functions,  144. 

Transcendental  function,  23,  300. 

Transcendental     functions,     theorems 
concerning,  301,  313,  317,  323. 

Transcendental  integral  function,  300, 
301. 

Transcendental     fractional     function, 
302,  317. 

Translation,  159. 

Transformation,  linear  fractional,  de- 
fined, 156. 
w  =  z-\-^,  159. 
w  =  az,  159. 
w=  az-\-  P,  162. 

1 
u)  =  -,  166. 


Upper  Umit,  28,  39,  40. 
Uniform  continuity,  35. 
Uniform  convergence  of  function  along 
a  boundary,  37. 
consequence  of,  38,  71,  75,  255. 
Uniform  convergence  of  series,  217. 
condition  for,  220. 

consequence  of,  221,  223,  225,  235, 
236. 

Velocity-potential,  98. 

Zero  point,  definition  of,  266,  360. 

order  of,  266,  360. 
Zero  points,  theorems  concerning,  267, 

269,  294,  296,  298,  301,  313,  325, 

335. 


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nen   einer   und    .ehrere    PComplexen 
-^  -I    - '  J  -i  - 

yxiisch-Functionentheoretische    Vor- 
-lesungen. 

ISxts 

?urkhardt-.Rasor    :    Theory  of    Functions    of    a 
Complex   Variaole, 

'lourt.t-Pedrick-Dunkel:  Functions    of    a   Cooiplex       ^ 
Variable. 

Pi-J-rpont    :    Functions   of    a   Complex    Variable. 
?iske    :    Functions   of   a   Co-nplex   Variable 
Harkness   and   I/oriey    :    Introduction    to    i^nalytic 

^'unctions 
Dur4ge:-Fisher   and    Schwatt:    Eleaients    of    the    Theory 

of    Functions  / 
KowaltiifsKi:    Die   Komplexen    Verainderlichen    und    ihre 
•Punctionen . 
(Written    fror    standpoint    of    applied    rcat hematics) 

Whittak^ir    :Modenn    Analysis. 

J/acHoDert    :    Functions   of    a   Coiplex    Vafiaole 

iscel Igneous  -  ,    / 

'or~l''7''Le(^ons   sur   les    Fonctions    Mono^enes    onifoVme 

d'une  Variable  Complexe. 

I  ••  Cours.' 


